Optical Interface Based on a Nanofiber Atom-Trap

Transcription

Optical Interface
Based on a Nanofiber
Atom-Trap
Dissertation
zur
Erlangung des Grades
Doktor
”
der Naturwissenschaften“
am Fachbereich Physik
der
Johannes Gutenberg-Universität
in Mainz
Eugen Vetsch
aus Mainz
Mainz, 2010
D77
1. Gutachter:
2. Gutachter:
Tag der mündlichen Prüfung: 12.04.2011
Abstract
In this thesis, I present the realization of a fiber-optical interface using optically
trapped cesium atoms, which is an efficient tool for coupling light and atoms. The
basic principle of the presented scheme relies on the trapping of neutral cesium
atoms in a two-color evanescent field surrounding a nanofiber. The strong confinement of the fiber guided light, which also protrudes outside the nanofiber, provides
strong confinement of the atoms as well as efficient coupling to near-resonant light
propagating through the fiber.
In chapter 1, the necessary physical and mathematical background describing the
propagation of light in an optical fiber is presented. The exact solution of Maxwell’s
equations allows us to model fiber-guided light fields which give rise to the trapping
potentials and the atom-light coupling in the close vicinity of a nanofiber. In addition, the fabrication of tapered optical fibers (TOFs) with a nanofiber waist from
standard glass fibers is presented. Chapter 2 gives the theoretical background of
light-atom interaction. A quantum mechanical model of the light-induced shifts of
the relevant atomic levels is reviewed, which allows us to quantify the perturbation
of the atomic states due to the presence of the trapping light-fields.
The experimental realization of the fiber-based atom trap is the focus of chapter 3.
Here, I analyze the properties of the trap in terms of the confinement of the atoms
and the impact of several heating mechanisms. In particular, the detrimental effect
of fundamental and technical noise on the storage time due to heating and subsequent loss of atoms is examined. Furthermore, I demonstrate the transportation
of the trapped atoms, as a first step towards a deterministic delivery of individual
atoms.
In chapter 4, I present the successful interfacing of the trapped atomic ensemble and
fiber-guided light. Three different approaches are discussed, i.e., those involving the
measurement of either near-resonant scattering in absorption or the emission into
the guided mode of the nanofiber. In the analysis of the spectroscopic properties
of the trapped ensemble we find good agreement with the prediction of theoretical
model discussed in chapter 2. In addition, I introduce a non-destructive scheme for
the interrogation of the atoms states, which is sensitive to phase shifts of far-detuned
fiber-guided light interacting with the trapped atoms. The inherent birefringence
in our system, induced by the atoms, changes the state of polarization of the probe
light and can be thus detected via a Stokes vector measurement.
Contents
Introduction
1
1 Light propagation in step-index fibers
5
1.1
Maxwell’s equations in a step-index optical fiber . . . . . . . . . . .
1.2
Solution for the fields
. . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.3
Field distribution of the HE11 mode in and outside a nanofiber . . .
14
1.3.1
HE11 mode with rotating polarization . . . . . . . . . . . . .
14
1.3.2
HE11 mode with quasi-linear polarization . . . . . . . . . . .
20
Tapered optical fibers (TOFs) . . . . . . . . . . . . . . . . . . . . . .
26
1.4
2 Atom-field interaction
2.1
6
31
Lorentz model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
2.1.1
Scattering rate . . . . . . . . . . . . . . . . . . . . . . . . . .
33
2.1.2
Dipole potential . . . . . . . . . . . . . . . . . . . . . . . . .
34
2.1.3
Multilevel atoms . . . . . . . . . . . . . . . . . . . . . . . . .
35
2.2
AC-Stark shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
2.3
Light shifts and potentials in a two-color dipole trap . . . . . . . . .
43
3 Trapping of neutral atoms
3.1
3.2
49
MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
3.1.1
. . . . . . . . . .
51
Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
Doppler cooling/Velocity dependent forces
. . . . . . . . . . . . . . . . . . . . . .
54
. . . . . . . . . . . . . . . . . . . . . . . . . .
55
Magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . .
56
Nanofiber trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
3.3.1
Loading of the trap . . . . . . . . . . . . . . . . . . . . . . . .
60
3.3.2
Atom number measurement . . . . . . . . . . . . . . . . . . .
63
3.3.3
Polarizations of the evanescent fields . . . . . . . . . . . . . .
64
3.3.4
Trap depth and -frequencies . . . . . . . . . . . . . . . . . . .
66
3.3.5
Trap lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
3.4
Temperature measurement . . . . . . . . . . . . . . . . . . . . . . . .
71
3.5
Heating-rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
3.5.1
Recoil heating
. . . . . . . . . . . . . . . . . . . . . . . . . .
77
3.5.2
Heating due to dipole-force fluctuations . . . . . . . . . . . .
78
3.5.3
Potential fluctuations . . . . . . . . . . . . . . . . . . . . . .
79
3.5.4
Influence of phase noise . . . . . . . . . . . . . . . . . . . . .
83
Conveying atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
3.6.1
Moving standing wave . . . . . . . . . . . . . . . . . . . . . .
87
3.6.2
Transport of atoms . . . . . . . . . . . . . . . . . . . . . . . .
89
Imaging of the trapped atoms . . . . . . . . . . . . . . . . . . . . . .
89
3.7.1
Fluorescence imaging . . . . . . . . . . . . . . . . . . . . . . .
89
3.7.2
Coherent scattering . . . . . . . . . . . . . . . . . . . . . . .
92
3.3
3.6
3.7
3.2.1
Cooling-Laser system
3.2.2
Vacuum setup
3.2.3
4 Interfacing light and atoms
4.1
95
Resonant interaction . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
4.1.1
Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
4.1.2
Optical pumping . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.2
Fluorescence of the atomic ensemble . . . . . . . . . . . . . . . . . . 103
4.3
Nondestructive Phase Detection . . . . . . . . . . . . . . . . . . . . . 107
5 Conclusions and Outlook
5.1
117
Coherent light-matter interaction . . . . . . . . . . . . . . . . . . . . 119
5.1.1
EIT and quantum memories . . . . . . . . . . . . . . . . . . . 119
5.1.2
Dark state polariton physics . . . . . . . . . . . . . . . . . . . 121
List of Figures
125
List of Tables
127
Bibliography
129
Introduction
Since the first exploration of individual quantum systems, a large variety of experimental work has been carried out in order to investigate quantum effects with the
aim of testing several models and predictions of quantum theory [1]. The feasibility
to efficiently access and manipulate the quantum degrees of freedom of such systems has even initiated research on quantum information processing [2]. For this
purpose many different approaches (employing, e.g., ions, neutral atoms, quantum
dots, and other solid state devices [7, 9] as quantum systems) have been proposed,
and have been, to varying degrees, successfully demonstrated.
In view of the requirements for practical applications of, e.g., quantum information technologies [3], neutral atoms are one of the most promising candidates as
storage units for quantum information (qubits) [5, 6]. Due to the weak coupling of
neutral atoms to their environment, qubits encoded as coherent superposition of
internal atomic states, can be robust against external influences, which means that
decoherence, i.e., loss of quantum information can be minimized. Moreover, recent
advances in atomic/optical physics allows for the coherent control and manipulation
of atomic external and internal states [6].
At the same time, solid state quantum devices, such as quantum dots or superconducting circuits, are readily miniaturized and integrated using well established
technologies [7]. For these reasons, the possibility of combining the advantageous
properties of atomic and solid state devices in hybrid quantum systems has recently
attracted considerable interest [8–10]. In order to realize such a hybrid quantum
system, the atoms need to be efficiently interfaced with near-resonant probe light for
manipulation and interrogation. Moreover, the atoms need to be trapped in close
vicinity of the solid state devices in order to be coupled via electric or magnetic
interaction.
Ideally, the optical interface would be mediated by fiber-guided light which could
bridge efficiently long distances. In this respect, using tapered optical fibers (TOF)
provides both prerequisites, efficient light-atom coupling, and low-loss transmission
of the light. These TOF feature a highly efficient coupling of the fiber-guided
mode in and out of the nanofiber waist [11, 12], resulting in a strongly confined
optical mode which couples to the atoms via the evanescent field [15, 32]. In this
case, the atomic ensembles can be directly interfaced with the fiber-guided light,
making the quantum fields intrinsically mode-matched and efficiently coupled into
single-mode optical fiber [16, 17]. The realization of such an interface between
different quantum systems may not only provide key technological requirements for
quantum information processing, but it also bares potential for the implementation
of quantum communication schemes over long distances using optical fibers.
The transmission of bits of data on the level of single photons in combination
with quantum cryptography techniques promises a provably secure way to exchange
information [18, 19]. In regard to the quantum nature of photons they can be
prepared in pairs of non-locally connected (entangled) states carrying an encoded
qubit, which forms the basis for most optical quantum communication schemes [20].
Using optical fibers as transmission channels for these entangled photons would
appear to be the best option, simply because of the high bandwidth and low losses
provided by optical glass fibers over long distances. For the same reasons, classical
telecommunication is mainly based on optical fiber networks.
Fiber-based quantum communication schemes have been realized by a few research
groups, demonstrating quantum key distribution over hundred kilometers [21, 22].
However, their applications in large-scale (global) networks is not straightforwardly
possible. Due to losses and decoherence of the quantum channels, the communication fidelity decreases exponentially with the fiber length, making direct quantum
communication techniques impractical for distances much larger than 100 km [9].
This limitation can be overcome by subdividing the long distances to be bridged into
smaller segments. The entanglement can be extended by connecting the adjacent
segments via entanglement swapping in each section [20,23]. Because the nature of
this approach is probabilistic, it requires storing the successfully entangled segment
state in a quantum memory, while waiting for the others to be generated. It can be
shown that the number of required segments as well as the communication time rises
only polynomially with growing distances [20]. A key element of such a “repeater”
approach is the transfer of quantum information between (“flying”) photonic qubits
and stationary matter-based qubits providing low decoherence rates [20, 24, 25].
Most recently, quantum repeaters as well as quantum memories based on the coupling of freely propagating light beams with laser-cooled atomic ensembles have
been successfully demonstrated [26, 27]. In these experiments, the entanglement,
encoded in collective single spin excitations (spin waves), was stored in the ensemble and could subsequently transferred to single photons. This approach takes
advantage of the collective enhancement of light-matter interaction (super radiance), making the transfer of quantum information very efficient. However, the
dephasing of the spin wave due to thermal atomic motion limits the storage time.
Furthermore, the mode-matching and fiber coupling with free beam optics limits the
photon collection efficiency. Introducing fiber-coupled and trapped atoms instead
might help to overcome these issues.
The coupling of laser-cooled atoms with fiber-guided light has been an active field
of research over the past few years. For this purpose, two types of optical fibers
have been considered: hollow core fibers and optical nanofibers. In the former, the
atoms are funneled into a capillary in the center of the fiber where they couple to
the guided fiber mode [28, 29], while in the latter, the atoms remain at the outside
of a nanofiber with a diameter smaller than the wavelength of the guided light and
couple to the evanescent field surrounding the fiber [15, 16]. The ultimate goal in
both lines of research is to combine the coupling scheme with three dimensional
trapping of the atoms in order to avoid dephasing due to thermal motion and
collisions, thereby maximizing both the number of coupled atoms, as well as the
optical density and the interaction time. In this context, it has been proposed
to realize a two-color optical dipole trap which makes use of a red-detuned and
blue-detuned evanescent light field around the optical nanofiber [30, 31].
In this thesis, I demonstrate the realization of an optical interface with trapped neutral cesium atoms carried by a two-color evanescent field surrounding the nanofiber
waist of a TOF. This work is a first step towards fiber-based quantum memories
and quantum hybrid systems. The atoms are localized in a one-dimensional optical
lattice about 230 nm above the nanofiber surface and can be efficiently interfaced
with a resonant light field sent through the nanofiber. Strong confinement of the
atoms in all three dimensions is achieved by a periodic standing wave potential in
conjunction with a azimuthally varying intensity distribution for the linearly polarized evanescent field. Due to the small trapping volumes the loading operates
in the collisional blockade regime resulting in an occupancy of at most one atom
per trapping site. Each of the trapped atoms absorbs up to 1.6% of the resonant
propagating light power. In conjunction with the 2000 trapped atoms this yields an
optical density (OD) of 32. Such strong coupling of the atoms and the fiber-guided
modes even allows us to interrogate the atoms non-destructively by means of an
off-resonant phase measurement.
Publications related to this thesis:
• E. Vetsch, D. Reitz, G. Sagué, R. Schmidt,S. T. Dawkins, and A. Rauschenbeutel, Optical Interface Created by Laser-Cooled Atoms Trapped in the Evanescent Field Surrounding an Optical Nanofiber, Phys. Rev. Lett. 104, 203603
(2010).
• S. T. Dawkins, E. Vetsch, D. Reitz, R. Mitsch, and A. Rauschenbeutel, Nondestructive Phase Detection with fiber coupled atoms, In preparation (2010).
• G. Sagué, E. Vetsch, W. Alt, D. Meschede and A. Rauschenbeutel, Cold
Atom Physics Using Ultra-Thin Optical Fibers: Light-Induced Dipole Forces
and Surface Interactions, Phys. Rev. Lett. 99, 163602 (2007).
4
Introduction
• F. Warken, E. Vetsch, D. Meschede, M. Sokolowski, and A. Rauschenbeutel,
Ultra-sensitive surface absorption spectroscopy using sub-wavelength diameter
optical fibers, Optics Express 15, 19 11952-11958 (2007).
Chapter 1
Light propagation in step-index
fibers
Standard optical glass fibers, used in telecommunication or for scientific purposes,
are based on a cladded-core structure (see Fig. 1.1). Core and cladding are produced
by chemical vapor deposition (CVD) of SiO2 doped with germanium or fluorine,
respectively, in order to obtain a slightly higher refractive index in the core. Due
to the higher index of refraction, light is confined to the core and can be guided by
total internal reflection along the fiber.
However, since the difference in the refractive index is less than 1%, the propagating
mode leaks far out of the core in the form of the evanescent field and co-propagates
in the cladding region. A thick layer of glass cladding is therefore required to isolate
the guided light from the outside. Typical dimensions of the fibers are 2-5 µm-radius
of the core surrounded by 60-µm glass cladding. The existence of the evanescent
field, which is a consequence of the boundary conditions for electromagnetic waves
at the core-cladding interface, is the basis for many applications employing aircladded fibers. Chemical sensing [35, 36], light splitting and combining, etc. are
some examples.
Recently, much attention has been given to sub-wavelength diameter tapered optical
fibers (TOFs). These nanofibers feature a strong transverse confinement of the
guided fiber mode. In addition, the projection of the mode outside the core results
in an intense evanescent field surrounding the fiber. In the following, the theoretical
concepts of guiding light in nanofiber waveguides and the distribution of fields
outside the nanofiber, which is the basis of the current work, will be given.
6
Light propagation in step-index fibers
1.1
Maxwell’s equations in a step-index optical fiber
In this section, a brief introduction to the description of light propagation and the
field distribution in and outside an optical fiber, will be given. The exact solution
of Maxwell’s equations is presented, thus providing the foundation of modeling the
trapping potential as well as the coupling of the evanescent field to the atoms. A
detailed treatment of these issues can be found in [32–34] and is closely followed
here.
b)
a)
2b
2a
φ
r
r
n(r)
z
n1
a
n2
n(r) =
n1 if r < a
with n1 > n2
n2 if r > a
(1.1)
Figure 1.1: a) Geometry of a standard glass fiber and b) the profile of its
refractive index as a function of r.
A schematic of the structure of a step-index cylindrical fiber is shown in Fig. 1.1.
Note that for a nanofiber the cladding is replaced by vacuum. The refractive index
n as a function of r is given by Eq. (1.1), where a denotes the core radius, n1 the
refractive index inside the fiber, and n2 the refractive index in the surrounding
medium (cladding). The wave equation of light propagation in such a fiber [33],
2~
~ 2E
~ − µ0 ε(r) ∂ E
∇
∂t2
~
= −∇(
~
E
~
· ∇ε(r))
ε(r)
(1.2)
~ ·H
~ =0
∇
(1.3)
can be derived from Maxwell’s equations:
~
~ ×H
~ = ε(r) ∂ E ,
∇
∂t
~
~ ×E
~ = −µ0 ∂ H ,
∇
∂t
~ · (ε(r)E)
~ =0
∇
1.1 Maxwell’s equations in a step-index optical fiber
7
~ denotes the electric field vector, µ0 the vacuum permeability, and ε(= n2 )
Here, E
the electric permittivity of the medium. The wave equation for the magnetic field
~ takes the same form as Eq. (1.2). For a cylindrical symmetric wave guide, e.g.
H
a circular step-index fiber, the right hand side of Eq. (1.2) vanishes. Due to this
symmetry, it is convenient to derive the field components in cylindrical coordinates
(r, φ, z). The solutions of Eq. (1.2) for the axial component take the simple form
of
Ez (~r, t)
Ez (r, φ)
=
exp[i(ωt − βz)],
(1.4)
Hz (~r, t)
Hz (r, φ)
where β is the axial propagation constant of the field. Moreover, employing Maxwell’s
equations in cylindrical coordinates:
iωεEr
iωεEφ
∂
∂
= iβ ∂r
Ez + 1r ∂φ
Hz
∂
= −iβHr − ∂r Hz
−iωµHr =
−iωµHφ =
(1.5)
∂
iβ ∂r
Hz
∂
+ 1r ∂φ
Ez
∂
−iβEr − ∂r Ez
all remaining components, Er , Hr , Eφ and Hφ , can be expressed in terms of Ez
and Hz .
−iβ
ωµ ∂
∂
Er = ω2 µε−β
E
+
H
z
2
β r∂φ z ∂r
−iβ
∂
∂
Eφ = ω2 µε−β 2 r∂φ Ez − ωµ
β ∂r Hz
Hr
=
Hφ =
(1.6)
−iβ
∂
ωε ∂
ω 2 µε−β 2 ∂r Hz − β r∂φ Ez
−iβ
∂
∂
H + ωε
β ∂r Ez .
ω 2 µε−β 2 r∂φ z
The solution of the whole problem can now be reduced to the solution of the wave
equation for the z-components of Eq. (1.2)
h
i E (r, φ) 1
1
z
∂r2 + ∂r + 2 ∂φ2 + (k2 − β 2 )
= 0,
(1.7)
Hz (r, φ)
r
r
where k2 = µ0 εω 2 .
This equation is separable with solutions of the form
Ez (r, φ)
= R(r) exp[±ilφ],
Hz (r, φ)
(1.8)
where l = 0, 1, 2, ...
For the radial functions, R(r), Eq. (1.7) becomes the Bessel differential equation
h
1
l2 i
∂r2 + ∂r + (k2 − β 2 − 2 ) R(r) = 0.
r
r
(1.9)
8
Figure 1.2: Plot of the Bessel functions of the first and second kind (J0 , Y0 ,
left), and the modified Bessel functions (I0 , K0 , right) all of order l = 0.
The general solutions of Eq. (1.9) are Bessel functions of order l depending on the
sign of k2 − β 2
R(r) = c1 Jl (hr) + c2 Yl (hr) for k2 − β 2 > 0, with h2 = k2 − β 2 ,
(1.10)
R(r) = c3 Il (qr) + c4 Kl (qr) for k2 − β 2 < 0, with q 2 = β 2 − k2 .
(1.11)
Here, Jl denotes the Bessel function of the first kind, Yl the Bessel function of the
second kind, Il the modified Bessel function of the first kind and Kl the modified
Bessel function of the second kind, all of order l.
The condition that any lossless modes be confined to the core, restricts the axial
propagation constant β within the range of values given by
n2 k0 ≤ β ≤ n1 k0 ,
(1.12)
where k0 = ω/c is the wave vector of the propagating field in vacuum. The solution
of Eq. (1.9) will therefore take the form of Eq. (1.10) inside the core, for r < a, and
the form of Eq. (1.11) in the surrounding medium, for r > a.
Since Yl is singular at r = 0 (compare Fig. (1.2)), the fields in Eq. (1.9) are only
finite if c2 = 0. For a lossless mode the power density is restricted to the fiber, the
guided fields must therefore vanish for large r.
Because Il diverges as r → ∞ (compare Fig. (1.2)), it requires c3 = 0 in order to
show an evanescent decay of the field outside the fiber.
The field components Ez and Hz can now be expressed as
Ez (r, φ, z, t) = AJl (hr) exp[i(ωt ± lφ − βz)],
Hz (r, φ, z, t) = BJl (hr) exp[i(ωt ± lφ − βz)],
q
with h =
n21 k02 − β 2
inside the core (r < a), and
(1.13)
(1.14)
(1.15)
1.1 Maxwell’s equations in a step-index optical fiber
Ez (r, φ, z, t) = CKl (qr) exp[i(ωt ± lφ − βz)],
Hz (r, φ, z, t) = DKl (qr) exp[i(ωt ± lφ − βz)],
q
with q =
β 2 − n22 k02
9
(1.16)
(1.17)
(1.18)
outside the core (r > a).
Using the expressions above together with Eqns. (1.3), (1.4) and (1.6), Er , Eφ , Hr
and Hφ are expressed in terms of Ez and Hz .
For r < a
βh
ωµ0 (±l) Jl (hr) i
iAJl′ (hr) −
B
exp[i(ωt ± lφ − βz)],
h
β
hr
i
βh
Jl (hr) iωµ0
(±l)A
+
BJl′ (hr) exp[i(ωt ± lφ − βz)],
h
hr
β
Er (r, φ, z, t) = −
Eφ (r, φ, z, t) =
Ez (r, φ, z, t) = AJl (hr) exp[i(ωt ± lφ − βz)],
(1.19)
βh
ωε1 (±l) Jl (hr) i
iBJl′ (hr) +
A
h
β
hr
i
Jl (hr) iωε1
βh
′
(±l)B
−
AJl (hr) exp[i(ωt ± lφ − βz)],
h
hr
β
Hr (r, φ, z, t) = −
Hφ (r, φ, z, t) =
Hz (r, φ, z, t) = BJl (hr) exp[i(ωt ± lφ − βz)],
(1.20)
where Jl′ (hr) = dJ(hr)/d(hr) and ε1 = n21 ε0 is the dielectric constant inside the
core or inside the nanofiber.
For r > a
βh
ωµ0 (±l) Kl (qr) i
iCKl′ (qr) −
D
q
β
qr
i
βh
Kl (qr) iωµ0
Eφ (r, φ, z, t) = − (±l)C
+
DKl′ (qr) exp[i(ωt ± lφ − βz)],
q
qr
β
Er (r, φ, z, t) =
Ez (r, φ, z, t) = CKl (qr) exp[i(ωt ± lφ − βz)],
(1.21)
βh
ωε2 (±l) Kl (qr) i
iDKl′ (qr) +
C
q
β
qr
i
βh
Kl (qr) iωε2
Hφ (r, φ, z, t) = − (±l)D
−
CKl′ (qr) exp[i(ωt ± lφ − βz)],
q
qr
β
Hr (r, φ, z, t) =
Hz (r, φ, z, t) = DKl (qr) exp[i(ωt ± lφ − βz)],
(1.22)
where Kl′ (qr) = dK(qr)/d(qr) and ε2 = n22 ε0 denotes the dielectric constant in the
cladding or in the surrounding medium.
10
The normalization constants A, B, C, and D as well as β, can be obtained by
considering the boundary conditions. The tangential components Eφ,z , Hφ,z have
to be continuous at the core-cladding boundary
Eφ,z (r = a)|core = Eφ,z (r = a)|cladding ,
Hφ,z (r = a)|core = Hφ,z (r = a)|cladding .
(1.23)
These considerations, together with Eqns. (1.19)–(1.22) yield the following relations
for the constants A B, C and D
AJl (ha) − CKl (qa)
i
h ωµ
i
h i(±l)
i
h ωµ
i
h i(±l)
0 ′
0 ′
Jl (ha) + C
K
(qa)
+
D
−
K
(qa)
A 2 Jl (ha) + B −
l
h a
hβ
q2a
qβ l
BJl (ha) − DKl (qa)
h ωε
i
h i(±l)
i
h ωε
i
h i(±l)
i
1 ′
2 ′
Jl (ha) + B
J
K
K
A
(ha)
+
C
(qa)
+
D
(qa)
l
l
hβ
h2 a
qβ l
q2a
= 0,
= 0,
= 0,
= 0.
(1.24)
Equations (1.24) lead to a non-trivial solution provided that the determinant of
their coefficients vanishes [34]. This requirement leads to the mode condition that
determines the propagation constant β of each mode
J ′ (ha)
h 1 2 1 2 i2 β 2
Kl′ (qa) n21 Jl′ (ha) n22 Kl′ (qa) l
+
,
+
+
= l2
haJl (ha) qaKl (qa) haJl (ha) qaKl (qa)
ha
qa
k0
(1.25)
Employing the solution for β, the following relations between the constants A, B,
C and D, which determine the strength of the field components, can be found
C
A
B
A
D
A
=
=
=
Jl (ha)
,
Kl (qa)
Kl′ (qa) −1
iβ(±l) 1
1 Jl′ (ha)
+
+
,
ωµ0 h2 a2 q 2 a2 haJl (ha) qaKl (qa)
C B
.
A2
(1.26)
Since the field amplitudes are related to the power of the electromagnetic field, A
can be determined by accounting for the energy flux in z-direction via
ω
Pz =
2π
Z
0
2π
ω
dt
Z
Sz dS,
S
(1.27)
1.2 Solution for the fields
11
where Pz is the total power transmitted through the fiber and Sz is the component
of the Poynting vector along the fiber axis (see Sect. 1.3.1).
Note that the ratio B/A = Er /Eφ is complex accounting for a relative phase of
π/2 between Er and Eφ and thus for circular polarization of the transverse field.
The term ±l in the equations above stems from the ansatz functions Eqns. (1.8) for
the solution of the wave function in cylindrical coordinates. The + (−) sign refers
to the solution with right-handed (left-handed) circulation of the transverse field
~ ⊥ = (Ex , Ey ) around the fiber axis. The solution for linearly polarized light can
E
be composed as a superposition of left- and righthanded circular fields, i.e.
1
Ezlin = √ (Ez+ + Ez− )
2
(1.28)
or can be obtained by using another type of ansatz function
Ez (r, φ) = R(r) cos[lφ] or Ez (r, φ) = R(r) sin[lφ].
(1.29)
In the next section the solutions of Eq. (1.25) for the propagation constants β and
the associated modes will be discussed. These results will be used to derive the
field intensity distribution and polarization orientation in the fundamental mode of
an optical nanofiber.
1.2
Solution for the fields
The equation (1.25), which result from the boundary conditions, expresses the
confinement of light in an optical fiber. The solution yields a discrete set of values
for β, each of which represents a different propagation mode in the fiber. This result
arises in general when treating confined waves in one or more dimensions, like e.g.,
the solutions of a particle confined in a potential well in quantum mechanics.
Rearranging equation (1.25), and making use of the relations
Jl′ (x) = Jl−1 (x) −
l
Jl (x),
x
1
Kl′ (x) = − [Kl−1 (x) + Kl+1 (x)],
2
(1.30)
yields the two following sets of solutions
n2 + n2 K (qa) + K (qa)
Jl−1 (ha)
l
l−1
l+1
1
2
=
+
+ R,
2
haJl (ha)
2qaKl (qa)
(ha)2
2n1
n2 + n2 K (qa) + K (qa)
Jl−1 (ha)
l
l−1
l+1
1
2
=
+
− R,
2
haJl (ha)
2qaKl (qa)
(ha)2
2n1
(1.31)
(1.32)
12
HE13
HE11
1/2
7,016
3,832
10,173
13,323
ha
HE12
V
1/2
3,832
7,016
10,173
13,323
ha
EH11
V
EH12
Figure 1.3: Plot of the right hand side (red) and left hand side (green) of
equations (1.31) (left plot) and (1.32) (right plot) for l = 1 over the argument
ha. The dashed lines mark the cut-off values of V . The intersections are solutions
corresponding to either EH1m or HE1m modes.
with
+ Kl+1 (qa) 2 lβ 2 1
1 2 i1/2
+
+
.
2qaKl (qa)
n1 k0
(qa)2 (ha)2
2n21
(1.33)
The +(−) sign in Eq. (1.31 & 1.32) denotes the two sets of solutions, the HE (−)
and EH (+) modes, representing the roots of equation (1.25) which is quadratic in
Jl′ (ha)/haJl (ha). The designation of the modes is based on the contribution of Ez
and Hz to the mode: Ez is larger (smaller) than Hz for the EH (HE) modes [34].
The equations 1.31 & 1.31 are transcendental in ha as well and can only be solved
graphically or rather numerically by plotting each side as a function of ha using
R=
h n2 − n2 2 K
1
2
l−1 (qa)
(qa)2 = a2 (n21 − n22 )k02 − (ha)2 .
(1.34)
Figure 1.3 shows the graphical solution of Eq. (1.31 & 1.32) for the EH1m (HE1m )
modes. The left- and right-hand sides of Eq. (1.31 & 1.32) have been plotted as a
function of the parameter ha for l = 1. Each crossing point represents one propagating mode in the fiber. The corresponding propagation constant β is determined
by (ha)× , where both curves cross, via
r
(ha)×
β = n21 k02 −
.
(1.35)
a
Each set of modes exhibits different solutions depending on the value l. These
modes are labeled as EHlm and HElm , where m denotes the different solutions of
Eq. (1.31 & 1.32) for a fixed l. For the two special cases with l = 0 a differentiated
nomenclature is used: EH0m alias TM, and HE0m alias TE. The notation TE
and (TM) refers to the vanishing Ez (Hz ) components of the transverse electric
(transverse magnetic) modes. Whereas the hybrid modes EH and HE have six non
vanishing field components, the TM and TE modes have only three [39].
1.2 Solution for the fields
13
TM01
1/2
2,405
8,654
5,520
V
TE01
11,792
ha
2,405
-1/2
TM02
5,520
V
8,654
11,792
ha
TE02
Figure 1.4: Plot of the right hand side (red) and left hand side (green) of
equations (1.31) (left plot) and (1.32) (right plot) for l = 0 over the argument ha.
The dashed lines mark the cut-off values of V . The intersections are the TE0m
and TM0m modes, respectively.
As pointed out in [39], in geometric optics hybrid modes can be interpreted in terms
of skew rays, i.e. circulating light trajectories around the fiber axis, whereas the
transverse modes TM and TE imply non-rotating ray trajectories, that lie in the
meridional plane. Hence, the number l can be associated with the z-component of
the orbital angular momentum of the propagating electromagnetic field [34, 45].
From the plots in Figs. 1.3 and 1.4 it is evident that for a given l, the number of
solutions is determined by ha = V , the position of the singularity on the righthand
side of Eq. (1.31). Here, V denotes the fundamental parameter for the fiber-field
system
q
V = k0 a n21 − n22 .
(1.36)
It can be shown that all modes (apart from HE11 ) have a cut-off value in V , which
can be found by the m roots of Jl (ha) = 0 (dashed vertical lines in Figure 1.3
and 1.4). At the cut-off, each propagation constant takes the value of β ≡ k0 n =
k0 n2 , where n = β/k0 is the mode index, i.e., the effective refractive index of the
corresponding mode, see Fig. 1.5. As a mode approaches cut-off, the fields penetrate
deeply into the cladding medium. Thus, the mode is poorly confined and poorly
guided, and most of the energy propagates in medium 2, leading to n = n2 . Similar
argument holds for far above cut-off, where the mode is tightly bound to the core
and thus n = n1 .
Below the cut-off value of V = 2.405 only the fundamental mode (HE11 ) can propagate. Hence, single mode condition is reached for a given wavelength λ, if the
radius of the fiber is small enough that only the fundamental mode can propagate.
For many applications requiring a well-defined phase front of the propagating light
(e.g. telecommunication, interferometry etc.), single-mode operation is compulsory.
In the present work, the light-induced trapping potential is formed by two fields of
14
n1
β/k
TE01
HE11
EH11
HE31
HE21
TM01
n2
0
1
3
2
4
HE12
5
V
Figure 1.5: Normalized propagation constant β/k as a function of V for a few
of the lowest-order modes of a step-index fiber [34], where k denotes the vacuum
wavenumber.
different wavelengths, both of which propagate in the fundamental mode HE11 of a
sub-wavelength optical fiber. Therefore, the analysis of the field distribution in the
next section covers only the HE11 mode. Note that it has been separately proposed
to realize a light-induced trapping potential which is based on the interference of
two low-order modes of a blue detuned evanescent light field [40]. Details of this
approach can be found in [39–41] and are not a part of this work.
1.3
Field distribution of the HE11 mode in and outside
a nanofiber
In this section the intensity distribution and polarization orientation of the fundamental HE11 mode of an optical nanofiber are discussed. The solutions of Maxwell’s
equations Eq. (1.10) permit the exact modeling of the fields inside and outside a
fiber with a diameter smaller than the wavelength of the guided light [32].
1.3.1
HE11 mode with rotating polarization
The field equations of the fundamental HE11 mode with rotating polarization in
~ in and outside the fiber is given by
terms of the electric field E
1.3 Field distribution of the HE11 mode in and outside a nanofiber
15
Figure 1.6:
Vectorial plot of the transverse field components
E~⊥ = (Ex (x, y), Ey (x, y)) at z = 0, for t = 0, t = π/4ω, t = π/2ω, t = 3π/4ω
respectively for a 250-nm radius vacuum-clad glass-fiber (n1 = 1.46) transmitting
σ + -polarized light at a wavelength of λ =852 nm.
β11
[(1 − s11 )J0 (h11 r) − (1 + s11 )J2 (h11 r)]
2h11
exp[i(ωt ± φ − β11 z)],
β11
Eφ (r, φ, z, t) = ± A
[(1 − s11 )J0 (h11 r) + (1 + s11 )J2 (h11 r)]
2h11
exp[i(ωt ± φ − β11 z)],
Er (r, φ, z, t) = − iA
Ez (r, φ, z, t) =AJ1 (h11 r) exp[i(ωt ± φ − β11 z)],
(1.37)
for r < a (inside the fiber), and
β11 J1 (h11 a)
[(1 − s11 )K0 (q11 r) + (1 + s11 )K2 (q11 r)]
2q11 K1 (q11 a)
exp[i(ωt ± φ − β11 z)],
β11 J1 (h11 a)
Eφ (r, φ, z, t) = ± A
[(1 − s11 )K0 (q11 r) − (1 + s11 )K2 (q11 r)]
2q11 K1 (q11 a)
exp[i(ωt ± φ − β11 z)],
J1 (h11 a)
Ez (r, φ, z, t) =A
K1 (q11 r) exp[i(ωt ± φ − β11 z)],
(1.38)
K1 (q11 a)
Er (r, φ, z, t) = − iA
for r > a (outside the fiber),
16
Figure 1.7: Ellipticity ε = |Er |/|Eφ | of the polarization of the electric fields
versus the radial distance r/a.
where
h11 =
q11 =
s11 =
q
q
h
2 ,
k02 n21 − β11
2 − k 2 n2 ,
β11
0 2
1
1 ih J1′ (h11 a)
K1′ (q11 a) i−1
+
+
.
(h11 a)2 (q11 a)2 h11 aJ1 (h11 a) q11 aK1 (q11 a)
(1.39)
Note that from the relation Er /|Er | = ±iEφ /|Eφ |, obtained from Eq.(1.37)–(1.38),
the radial and tangential components Er and Eφ have a phase difference of π/2.
Therefore, the orientation of the transverse field E~⊥ = (Er , Eφ ) rotates in time
along the fiber axis. In Fig. 1.6 the vectorial field plot of E~⊥ (t, z = 0) for different
times t shows the corresponding phases of rotation. However, this rotation is not
perfectly circular but elliptical and the ellipticity varies in space. Moreover, due to
~ in the transverse
the non-vanishing longitudinal component Ez , the rotation of E
plane is superimposed with an elliptical rotation in a plane parallel to the fiber
axis z. Thus, the normal vector of the polarization ellipse is not parallel but tilted
by the angle α = arctan (|Ez |/|Eφ |) with respect to the fiber axis. The ellipticity
ε = |Er |/|Eφ |, i.e., the aspect ratio of the major and minor axes, versus the radial
distance is shown in Fig. 1.7. Inside the core and far outside in the cladding
region the HE11 mode is almost purely circularly polarized. The deviation from
pure circular polarization, however, increases when approaching the surface of the
fiber. This behavior originates from the boundary conditions for the electric field
[32], which lead to a discontinuity in the Er component at the fiber surface, while
the tangential components Eφ and Ez remain continuous. Hence, this sets the
radial orientation of the ellipse, its major (minor) axis is directed perpendicular
(tangential) to the fiber surface.
Figure 1.8: Normalized intensities |E|2 in units of |E(r = a)|2 including the
cylindrical-coordinate components |Er |2 , |Eφ |2 , |Ez |2 of the cycle averaged squared
electric field modulus in the HE11 mode for rotating polarization, plotted in the
transverse plane (x, y).
17
18
(a)
(b)
2
E
Ei
2.0
2
1.0
1.5
0.8
0.6
1.0
0.4
0.5
0.2
0.0
0.5
1.0
1.5
r/ a
2.0
0.0
0.5
1.0
1.5
2.0
r/ a
Figure 1.9: Radial intensity distribution (a) |E(r)|2 and (b) |Ei (r)|2 of the
cylindrical-coordinate components of the HE11 mode for rotating polarization,
normalized by |E(r = a)|2 .
Intensity distribution
The boundary conditions for the normal component Er also lead to a discontinuity
of the intensity at the surface of the fiber. The large difference in refractive index
at the boundary of core and cladding causes a proportional jump in intensity, leading to an enhancement of the field strength outside the fiber. In the following the
intensity will be given in terms of the electric field via:
I=
1
~ 2.
cε0 |E|
2
(1.40)
2
~ denotes the squared modulus of the electric field averaged over one osHere, E
cillation period. Thus, the intensity distribution of the HE11 mode is given by
i
2 2 h
2
E(r)
~ 2 = A β11 (1 − s11 )2 J02 (h11 r) + (1 + s11 )2 J22 (h11 r) + 2 h11 J12 (h11 r) ,
2
in
2h211
β11
(1.41)
for r < a, and
E(r)
~ 2
out
=
i
2 J 2 (h a) h
2
A2 β11
q11
11
1
2 2
2 2
2
(1−s
)
K
(q
r)+(1+s
)
K
(q
r)+2
K
(q
r)
,
11
11
0 11
2 11
1 11
2 K 2 (q a)
2
2q11
β11
1 11
(1.42)
for r > a.
In
figure
1.8, the corresponding three dimensional plot of the intensity distribution
~ 2 of the HE11 mode for rotating polarization in the transverse plane is shown.
E(r)
This example refers to a 250-nm radius silica-fiber (n1 = 1.45), surrounded by
vacuum (n2 = 1) and transmitting circularly polarized light of 852-nm wavelength.
19
~ within one cycle, the cycle averaged squared modulus
Due to a full revolution of E
of the electric field, in Eq. (1.41) does not depend on the azimuthal coordinate φ.
The same holds true for the constituents of the total intensity |Er |2 , |Eφ |2 , |Ez |2 ,
whose angular dependence is averaged out within one oscillation period. Hence, the
intensity of the evanescent field in the HE11 mode, excited by circularly polarized
light, is distributed cylindrically symmetrically around the fiber.
Figure 1.9 shows the radial intensity distribution on the
inside and the outside of
~ 2 is normalized to maximal
a vacuum-clad fiber for the HE11 mode. Here, E(r)
~ (r = a) 2 outside the fiber. Inside the fiber, the intensity is maximal
intensity E
at the center of the fiber (r = 0) and decays towards the surface. At the fiber
surface, however, the intensity is enhanced due to the discontinuity of Er and decays
evanescently outside the fiber. The tangential components Eφ,z are continuous at
the surface. Their contributions to the total intensity outside the fiber are about
25% each. The decay length of the evanescent field, given by Λ = 1/q11 (Eq. 1.39)
scales with the wavelength of transmitted light. This wavelength dependency is
critical to the two-color trapping scheme and will be discussed in Sect. 2.3.
Power density
The normalization constant A of the field amplitudes in Eqs.(1.41 & 1.42) is related
to the total power of the light propagating in the fiber via the Poynting vector
i
h
~ = 1 Re E
~ × H~ ∗ .
hSi
2
(1.43)
The z-component of the cycle-averaged Poynting vector hSz i, quantifies the energy
flux of the electromagnetic field in the propagation direction, i.e., along the fiber
axis. Integration of hSz i over the transverse plane leads to the following expression
for the power propagating inside and outside the fiber [44]
Pin =
Pout =
Z
Z
2π
dφ
0
2π
dφ
0
Z
Z
a
0
hSz iin rdr,
∞
a
hSz iout rdr.
(1.44)
Using Eqns. (1.20), (1.22), and Eqns. (1.37), (1.38), the Poynting vector of the
electromagnetic field in the HE11 mode can be easily calculated. The resulting
relation between the total transmitted power P = Pin + Pout (Eq. (1.44)) and the
normalization constant A reads
s
−1/2
4µ0 ωP A=
D
+
D
,
(1.45)
in
out
πa2 β11
20
with
h
β2
Din = (1 − s11 )(1 + (1 − s11 ) 11
)(J 2 (h11 a) + J12 (h11 a))+
h211 0
i
β2
2
)(J
(h
a)
−
J
(h
a)J
(h
a))
,
+ (1 + s11 )(1 + (1 + s11 ) 11
11
1 11
3 11
h211 2
2
J12 (h11 a) h
β11
2
2
(1
−
s
)(1
−
(1
−
s
)
11
11
2 )(K0 (q11 a) − K1 (q11 a))+
K12 (q11 a)
q11
i
β2
+ (1 + s11 )(1 − (1 + s11 ) 211 )(K22 (q11 a) − K1 (q11 a)K3 (q11 a)) .
q11
(1.46)
Dout =
(1.47)
Where Din /(Din + Dout ) and Dout /(Din + Dout ) are the fractions of the power of
the fields that propagate inside and outside the fiber. P is the total power of the
transmitted light, which can be directly measured at the fiber output.
1.3.2
HE11 mode with quasi-linear polarization
From Eqns. (1.19 & 1.21) in Chapter 1.1, the solutions of the quasi-linearly polarized
fields can be expressed in terms of the superposition of two circular fields
(lin)
Ei
1
= √ (Ei+ + Ei− ), i ∈ (r, φ, z)
2
(1.48)
The +(−) sign in the equations above denotes the solution with right-handed (lefthanded) rotation of the polarization around the fiber axis, respectively. In cartesian
coordinates the fields become
β11
[(1 − s11 )J0 (h11 r) cos(ϕ0 )
2h11
− (1 + s11 )J2 (h11 r) cos(2φ − ϕ0 )] exp[i(ωt − β11 z)],
β11
Ey (r, φ, z, t) = Alin
[(1 − s11 )J0 (h11 r) sin(ϕ0 )
2h11
− (1 + s11 )J2 (h11 r) sin(2φ − ϕ0 )] exp[i(ωt − β11 z)],
Ex (r, φ, z, t) = Alin
Ez (r, φ, z, t) = iAlin J1 (h11 r) cos(φ − ϕ0 ) exp[i(ωt − β11 z)]
(1.49)
(a)
Ei
21
(b)
2
Ei
1.2
1.2
Ex 2
1.0
0.8
0.6
0.6
0.4
Ez 2
0.0
0.5
Ex 2
1.0
0.8
0.2
2
0.4
0.2
Ey 2
1.0
1.5
2.0
r/ a
0.0
Ey 2 Ez 2
0.5
r/ a
1.0
1.5
2.0
Figure 1.10: Radial intensity distribution |Ei (r)|2 of the carthesian-coordinate
components of the HE11 mode for linear x-polarization, normalized to
2
|E(r = a)| =1, (a) shown in (x, y = 0) direction, and (b) shown in (x = 0, y)
direction.
for r < a (inside the fiber), and
β11 J1 (h11 a)
[(1 − s11 )K0 (q11 r) cos(ϕ0 )
2q11 K1 (q11 a)
+ (1 + s11 )K2 (q11 r) cos(2φ − ϕ0 )] exp[i(ωt − β11 z)],
β11 J1 (h11 a)
Ey (r, φ, z, t) = Alin
[(1 − s11 )K0 (q11 r) sin(ϕ0 )
2q11 K1 (q11 a)
+ (1 + s11 )K2 (q11 r) sin(2φ − ϕ0 )] exp[i(ωt − β11 z)],
J1 (h11 a)
Ez (r, φ, z, t) = iAlin
K1 (q11 r) cos(φ − ϕ0 ) exp[i(ωt − β11 z)],
K1 (q11 a)
Ex (r, φ, z, t) = Alin
(1.50)
for r > a (outside the fiber).
Here, q11 , h11 and s11 are identical to those defined in Sec. 1.3.1. The normalization √
constant Alin is related to the maximum amplitude of the electric field via
Alin = 2A, with A defined for rotating polarization given in Sec. 1.3.1. The angle
ϕ0 describes the orientation of the transverse E~⊥ = (Ex , Ey ) field: ϕ0 = 0 leads to
polarization in the x-direction and ϕ0 = π/2 to polarization in the y-direction. The
designation quasi-linear polarization refers to the non-vanishing longitudinal component Ez of the HE11 mode. Note that Ez is out of phase by π/2 with respect to
the perpendicular components Ex and Ey . This phase difference causes an elliptical
~ field in a plane parallel to the fiber axis z. The magnitude of Ez
rotation of the E
varies radially as well as azimuthally as does the ellipticity. For ϕ0 − φ = 0, and π,
i.e., in the plane of polarization, Ez is substantial close to the fiber surface (see
Fig. 1.10). In the plane perpendicular to polarization (ϕ0 − φ = π/2 and 3π/2),
however, Ez is zero and hence the polarization is purely linear.
22
In Fig. 1.11(a) the transverse field vectors E~⊥ = (Ex , Ey ) at t = 0 and z = 0 as
functions of (x, y) are plotted. This example refers to a 250-nm radius vacuumclad (n2 = 1) glass-fiber (n1 = 1.46), transmitting light of 852 nm, polarized
in the x-direction. As can be seen in this plot, the electric field for the linearly
polarized HE11 mode is perpendicular to the fiber surface for φ = 0 and φ = π
and parallel for φ = π/2 and φ = 3π/2. However, due to the boundary conditions
~ the normal component is discontinuous at the fiber surface. Therefore, the
for E,
circular curvature of the surface of the fiber dictates a sinusoidal variation of the
polarization orientation angle θ = arctan[Ey /Ex ] with the azimuthal angle φ (see
Fig. 1.11(b). The deviations of the polarization orientation from x-direction can be
substantial for φ 6= n π2 , n = 0, 1, 2..., and are highest at the fiber surface. For an
increasing distance from the surface the deviation from x-polarization decreases.
As an example, at a distance of r = 2a from the fiber surface (i.e., at about the
trap distance) and φ = π/4 the ratio (Ey /Ex ) is about 0.1 yielding θ = 0.1/2π.
~ fields also result in an azimuthal dependency
The boundary conditions for the E
of the field distribution. Due to the discontinuity of the normal component at the
boundary, linear, i.e., non-rotating, polarization breaks the cylindrical symmetry of
~ ⊥ is normal and minimal
the system. Thus, the evanescent field is maximal where E
~ ⊥ is tangential to the fiber surface.
where E
Following from Eqs. (1.49) and (1.50) the square of the electric field modulus for
the HE11 mode averaged over one oscillation period is given by
2
2 h
2
E(r,
~ φ)2 = Alin β11 (1 − s11 )2 J 2 (h11 r) + (1 + s11 )2 J 2 (h11 r) + 2 h11 J 2 (h11 r)
0
2
2
2 1
in
4h11
β11
h2
i
2
+ 2 11
J
(h
r)
−
(1
+
s
)(1
−
s
)J
(h
r)J
(h
r)
cos[2(φ
−
ϕ
)]
,
11
11
11 0 11
2 11
0
2 1
β11
(1.51)
E(r,
~ φ)2
2
A2lin β11
J12 (h11 a)
2
out
4q11
K12 (q11 a)
h
q2 2
× (1 − s11 )2 K02 (q11 r) + (1 + s11 )2 K22 (q11 r) + 2 11
2 K1 (q11 r)
β11
q2
i
2
+ 2 11
K
(q
r)
+
(1
+
s
)(1
−
s
)K
(q
r)K
(q
r)
cos[2(φ
−
ϕ
)]
,
11
11
11
0
11
2
11
0
1
2
β11
(1.52)
=
(a)
23
(b)
Figure 1.11:
(a) Vectorial plot of the transverse field components
E~⊥ = (Ex (x, y), Ey (x, y)) at t = 0, z = 0 for a 250-nm radius vacuum-clad
glass-fiber (n1 = 1.46) transmitting 852-nm wavelength light polarized in the xdirection. (b) Polarization orientation angle as a function of the azimuthal angle
ϕ = arctan x/y for three different radial distances.
2
~ = |Ex |2 + |Ey |2 + |Ez |2 . The intensity distribution and the cartesian
where E
components |Ex,y,z (r, φ)|2 according to equation (1.49) and (1.50) for the HE11
mode with polarization in the x-direction (ϕ0 = 0) is shown in Fig. 1.12. Here, the
same parameters as in Sect. 1.3.1 are used. The maximum total intensity of the
evanescent field is found at φ = 0 and is composed of |Ex (r, 0)|2 + |Ez (r, 0)|2 , which
is apparent from the term cos[2(φ − ϕ0 )] in Eqs. (1.51) and (1.52). The minimum
total intensity, at φ = π/2, consists only of the x-component |Ex (r, π/2)|2 , which
is about 30% of the maximum value (compare Fig. 1.10).
In contrast to weakly focused and freely propagating light beams, the intensity as
well as the polarization of light in the HE11 mode varies azimuthally. Many applications involving light-atom interaction, however, demand a well-defined orientation
of the polarization, e.g., for the optical pumping of atoms. In view of these circumstances, a convenient approach may be to localize the atoms where the probing field
~ is parallel to the surface of the fiber and thus purely linearly polarized. As I will
E
show in Sect. 2.3, a linear array of trapping sites for atoms can be accomplished
by means of light-induced dipole forces and by utilizing the azimuthal dependency
of the evanescent field of the quasi-linearly polarized HE11 mode, which enables
strong azimuthal confinement of atoms trapped in the vicinity of the nanofiber.
24
Figure 1.12: Normalized intensities |E|2 in units of |E(x = a, y = 0)|2 including
of the Cartesian-coordinate components |Ex |2 , |Ey |2 , |Ez |2 of the cycle averaged
squared electric field modulus in the HE11 mode for quasi-linear polarization,
plotted in the transverse plane (x, y).
25
Figure 1.13: Normalized intensity of the evanescent field I(r = a)/P =
1/Aeff (r = a) in units of 1/σ = (3λ2 /2π)−1 as a function of the normalized fiber
radius a/λ. Calculated for circularly polarized light and assuming a refractive
index of 1.46. The resulting functional form holds universally for any λ.
Confinement of the fiber-guided light
In the upper examples, light is considered to be guided by a fiber with diameter 2a
which is smaller than the light wavelength. Such strong transversal confinement of
light results in a significant intensity of the field outside the fiber. By employing
the analysis from Sect. 1.1, it can be shown that the intensity of the evanescent field
increases for decreasing fiber diameter and takes a maximum value when a/λ = 0.23
(see Fig. 1.13) [36]. A further reduction of fiber radius makes the mode only weakly
guided, i.e., the mode area Aeff diverges and the intensity converges to zero. This
result holds universally for all wavelengths in the visible and near-infrared region,
i.e., where the refractive index of the glass fiber is about 1.46.
In the optimal case, the mode field area Aeff (= P/I) is a significant fraction of the
resonant absorption cross-section σ ≃ 3λ2 /2π of alkali-atoms. Hence, atoms in close
vicinity of an appropriate fiber can be efficiently coupled to the fiber-guided light via
the evanescent field. By the same argument, relatively low powers of in-coupled (far
off-resonant) dipole lasers are sufficient in order to form the light-induced potentials
necessary for trapping the atoms (see Sect. 2.3).
In the present work, a fiber with 250-nm radius is used in order that a/λ be close to
0.23 for the various wavelengths employed (1064, 852 and 780 nm), thus ensuring
single-mode operation and strong confinement.
26
1.4
Tapered optical fibers (TOFs)
The theoretical considerations of light propagation in an optical nanofiber have
been discussed in the previous section. Such nanofibers can be realized as the waist
of tapered optical fibers (TOFs) that optimally match the mode of a standard single
mode optical fiber with the fundamental nanofiber mode [37].
Figure 1.14: Schematic of a tapered optical fiber (TOF) showing the light propagation and conversion of the fundamental mode in the tapered region (a) to the
nanofiber waist (b) and back again to the fiber guided mode (c). The intensity
profiles of the guided modes are represented by the red curves.
Figure 1.14 shows a schematic of a tapered optical fiber (TOF) with a four-part
structure; the fiber waist (b) surrounded by two taper transitions (a) and (c) to
the original fiber. The intensity of the corresponding mode of light propagation is
represented by the red filled curves. In the unprocessed part of the single mode
TOF, the light is guided inside the core via total internal reflection at the core
cladding boundary. This propagation mode is often approximated by the weakly
guided and linearly polarized fundamental mode LP01 [34]. In the tapered sections
the weakly guided LP01 mode of the unstretched fiber is adiabatically transformed
into the strongly guided HE11 mode of the nanofiber and back to the LP01 mode on
the opposite side of the TOF [11]. The decreasing diameter of the core in the first
taper leads to a compression of the guided mode until the core radius becomes the
size of about the wavelength of the light, whereby the guided mode expands into the
cladding. In this case, the glass cladding acts as the new core and the surrounding
medium becomes the new cladding. Further decreasing the fiber diameter causes
the mode field to be more and more laterally confined until the light approaches
the waist. Thus, the intensity of the light field at the sub-wavelength diameter
waist is very high and is accessible from the outside via the evanescent field. At
the second taper transition the mode transformation is accomplished in reverse.
In order to minimize transmission loss, the taper transitions have to be carefully
shaped in order to prevent energy transfer to higher order modes in the cladding
which are not guided by either the nanofiber waist or the fiber core. In various
work, the adiabadicity criteria of taper transitions were analyzed [11–14], however,
the realization of highly efficient TOFs is technologically challenging.
1.4 Tapered optical fibers (TOFs)
27
Laminar Flow Box
burner
Figure 1.15: Schematic of the pulling rig [37]. A standard glass-fiber is clamped
between two stacked translation stages, which move and stretch the glass-fiber
above the flame of a stationary gas burner. The pulling procedure is performed
under clean room conditions in a laminar flow box.
Flame pulling of TOFs
The TOF being used throughout this work was produced in a computer-controlled
fiber pulling rig, which is described in detail in the Ph.D. thesis of F. Warken [37].
A standard optical glass fiber (Liekki 6/125 Passive) with 6 µm core- and 125 µm
outer diameter is tapered to a 500-nm diameter and 5-mm long waist, yielding an
overall transmission through the TOF of 97% (see Fig. 1.17).
The pulling process of this TOF can be briefly described as follows: The polymer
coating is removed and the glass fiber is carefully cleaned at the tapering position
before being fixed in the pulling rig. The pulling rig consists of two linear translation
stages, the translator and the stretcher, that allow sub-micrometer actuation of the
positioning with precisely aligned V-groove holders, on which the fiber is clamped
and tightened (see Fig. 1.15). An electronically controlled hydrogen-oxygen burner
is placed between these two holders, which provides a very clean and laminar flame.
The flame heats approximately 1 mm of the fiber to about 1500◦ C, making it soft
and malleable, while the translator moves the fiber back and forth above the flame
as the stretcher simultaneously elongates it. Due to volume conservation the fiber
diameter is thus gradually reduced. This so-called “flame brushing technique”
allows one to produce almost any profile and waist diameter of the TOFs [37].
The required trajectories of the traveling stages are calculated by simulating the
pulling process on a computer. In particular, linear conical tapers with a constant
slope were found empirically to be optimal for the fiber mentioned above. At the
crucial diameters, below 40 µm, where the core mode is transformed into a cladding
mode, a very shallow slope should ensure impedance matching, thus minimizing
transmission loses.
28
(a)
(b)
Figure 1.16: (a) Optical microscope picture of a TOF. (b) SEM picture of the
waist of a TOF. The diameter of the waist was measured to be 0.5 µm.
In order to find the best taper parameters laser light was launched into the fiber
and its transmission was monitored during the pulling process (see Fig.1.17). An
optimum slope angle was found at 1.5 mrad with transmission of better than 99%
for a 500-nm diameter waist [42], while keeping the length of the TOF reasonably
short (≈ 7 cm). Furthermore, by monitoring the Rayleigh scattered light from
the nanofiber, we have found that the polarization of the transmitted light is not
significantly changed in the taper transition and the nanofiber waist. We are even
able to precisely align the polarization of the fiber-guided light on the nanofiber
waist with a residual ellipticity of less than one percent for a non-polarization
maintaining fiber (see, Sect. 3.3.3).
Figure 1.16 shows exemplarily a micrograph of a taper transition (a), and a SEM
(Scanning Electron Microscope) picture of the waist of a TOF with a diameter of
500 nm (b) [37]. Several electron microscopic imaging have also been produced
[37,39], showing a reproducible and homogenous waist diameter as well as a surface
roughness below the resolution limit of a SEM (≈ 1 nm).
These unique physical properties in conjunction with a high transmission makes the
TOF ideally suited for realization of a two-color evanescent wave trap around the
sub-wavelength diameter waist. The most critical factor in this context turns out to
be the robustness against fusing in vacuum. However, for optimal conditions, i.e.,
a perfectly straight TOF, and gently stretched nanofiber, these TOFs are capable
to carry few hundreds of milliwatts of laser light in vacuum without showing any
degradation of the fiber.
1.4 Tapered optical fibers (TOFs)
29
rel. Transmission
1,0
0,8
0,6
0,4
0,2
0,0
0
5
10
15
20
25
Time (a.u.)
Figure 1.17: Transmission of a 780 nm (blue) and 850 nm (red) laser light
through a TOF during the pulling procedure as a function of time. The oscillations
of the transmission at 780 nm at the beginning are caused by interferences with
higher order modes. The final diameter of the fiber waist is 500 nm (t=22),
allowing single mode transmission of 97% for both wavelengths.
Chapter 2
Atom-field interaction
In order to exploit the unique quantum properties of neutral atoms, it is necessary
to both isolate them from the environment and to achieve coherent control of the
relevant internal and external degrees of freedom. Both of these steps involve the
understanding and utilization of the light-atom interaction.
This chapter gives a theoretical background of light and atom interactions, required
for the understanding of the principles of laser cooling and trapping. This treatment
will be based on the classical Lorentz model which describes the interaction of a
two-level atom with light in an intuitive way. Based on these results, the concepts
of atom trapping in a far off-resonant light field by means of light induced dipole
forces, will be extended to real world multilevel atoms.
Furthermore, a quantum mechanical model of the atomic eigenstates in the presence
of far-detuned light fields will complete the theoretical part, which will allow us to
estimate the effects of interrogation and manipulation of the atomic states in such
a perturbed system.
2.1
Lorentz model
The interaction of a monochromatic light field and a neutral two-level atom is
well described by a classical harmonic oscillator model (Lorentz model) [47]. In
this model the atom is reduced to an electron (mass me , charge −e) bound to
a massive core (mcore ≫ me , with charge +e) by a harmonic potential. In the
presence of a light field this electron is driven by the time varying electric field
E(t) = eE0 e−iωt +c.c., with the polarization vector e and the amplitude E0 , leading
to the equation of motion of a driven harmonic oscillator
ẍ(t) + Γω ẋ(t) + ω02 x(t) = −
e
E(t),
me
(2.1)
32
where the damping is described by the energy dissipation rate due to classical dipole
radiation [47]
e2 ω 2
Γω =
,
(2.2)
6πε0 me c3
and ω0 is the resonance angular frequency of the oscillator. The stationary solution
of Eq. (2.1)
α(ω)
E(t),
(2.3)
x(t) = −
e
yields an induced dipole moment of the atom d(t) = −ex(t) = α(ω)E(t), where
α(ω) =
1
e2
,
2
2
me ω0 − ω − iΓω ω
(2.4)
is the polarizability of the atom.
The polarizability α(ω) is a complex function of the frequency, where the real and
imaginary part describe the in-phase and the quadrature components of the atomic
dipole moment d, respectively.
By employing e2 /me = 6πε0 c3 Γω /ω 2 and the damping rate on resonance Γ ≡
Γω0 = (ω0 /ω)2 Γω the following relations for the phase and amplitude of the atomic
response, describing the dispersion and absorption, can be obtained
(ω02 − ω 2 )Γ/ω02
,
(ω02 − ω 2 )2 + Γ2 (ω 3 /ω02 )2
Γ2 ω 3 /ω04
Im[α(ω)] = 6πε0 c3 2
.
(ω0 − ω 2 )2 + Γ2 (ω 3 /ω02 )2
Re[α(ω)] = 6πε0 c3
(2.5)
(2.6)
Im[α(ω0 )]
Re[α(ω0 − Γ/2)]
(= Im[α(ω0 )]/2)
0
ω0 − Γ/2
ω0
ω0 + Γ/2
ω
Figure 2.1: Real (red) and imaginary (black) parts of the polarizability α(ω) as
a function of the angular frequency ω in the classical Lorentz model.
2.1 Lorentz model
2.1.1
33
Scattering rate
The time averaged power dissipation due to the damping rate Γ of a driven harmonic
oscillator, described by
D
E ω|E |2
0
P = Re[ḋ(t) · E(t)] =
Im[α(ω)],
(2.7)
2
shows that the quadrature part of the atomic dipole moment describes the absorption of energy of the light field. In a classical model, energy is dissipated
continuously, however in a quantum mechanical picture photons are emitted at the
transition from an excited state |ei to the ground state |gi with a scattering rate
P
|E0 |2
=
Im[α(ω)].
(2.8)
~ω
2~
Note that furthermore the classical model applies only for low intensities of the
driving field (I = |E0 |2 /2cǫ0 ), where the polarizability does not depend on I. However, in the near resonant and high intensity regime the ground and excited state
population of the atoms equilibrates, which leads to the saturation of the polarizability [5]
6πε0 c3
1
,
(2.9)
Im[α(ω, I)] =
3
ω0 1 + I/I0 + (2∆/Γ)2
Rs =
Re[α(ω, I)] =
6πε0 c3
2∆/Γ
,
3
ω0 1 + I/I0 + (2∆/Γ)2
(2.10)
where ∆ = ω − ω0 , Γ is the decay rate of the excited state or the full width at half
maximum (FWHM) of the transition line in units of rad/s, and
π~ω03
12πc2
is the saturation intensity. Thus, the scattering rate becomes
"
#−1
ΓI
I
2∆ 2
Rs (I, ∆) =
1+
+
.
2 I0
I0
Γ
I0 = Γ
(2.11)
(2.12)
At resonance (∆ = 0) and strong saturation (I ≫ I0 ), the scattering rate of single atoms can be substantial Γ/2 ≈ 107 photons/s. However, even at moderate
saturation in conjunction with strong transversal confinement of the probe light,
scattering results in a significant absorption [43]. Hence, the resonant light-atom
interaction can be very efficient for interrogation and manipulation of atomic states.
For detunings larger than Γ/2 the in-phase part of the atomic dipole moment becomes dominant (see Fig. 2.1), leading to a phase shift of the light with negligible
absorption (see Sect. 4.3). At the same time, the back action of the induced dipole
moment on the driving electric field results in dipole forces being exerted on the
atoms.
34
2.1.2
Dipole potential
The time-averaged interaction potential of the induced dipole moment d and the
(linearly polarized) driving electric field E is given by
1
1
U = − hd(t) · E(t)]i = − |E0 |2 Re[α(ω)].
2
4
(2.13)
Here, the first factor of − 12 accounts for the induced dipole moment and another
for the cycle average of squared electric field modulus.
1
2
For an atom (in the electronic ground state) subjected to a linearly polarized
light field Eq. (2.13) describes the light-induced dipole potential, which in a spatially varying light field E(r, t) gives rise to dipole forces exerted on the atom
F = −∇U (r). Considering the intensity of the field I(r) = |E0 (r)|2 /2cǫ0 and
Eq. (2.5) the following expression for the optical dipole potential is obtained
U (r, ∆) = −
1
I(r)Re[α(∆)].
2cǫ0
(2.14)
From the plot in Fig. 2.1 it is evident that for red detuning (∆ = ω − ω0 < 0)
the potential U (r) is attractive, i.e. the atom is attracted towards the intensity
maximum of the light field. Conversely, for blue detuning (∆ = ω − ω0 > 0) the
potential is repulsive, pushing the atom towards the intensity minimum. Strongly
focused laser beams offer large intensity gradients (∂i I(r)), making optical trapping
of dielectric particles (in, e.g., optical tweezers) with relatively low powers possible.
For large detunings of the trapping light, i.e., |ω0 −ω| ≫ Γ, the frequency dependent
part of the dipole potential and of the scattering rate reduces to
3πc2 IΓ
1
1
U =−
+
(2.15)
ω − ω0 ω + ω0
2ω03
2
3πc2 IΓ2 ω 3
1
1
Rs =
+
.
(2.16)
2~ω03 ω03 ω − ω0 ω + ω0
Using the rotating wave approximation and dropping fast oscillating terms, this
leads to an expression for the rate of off-resonant scattering induced by the dipole
fields
Γ
Rs =
U.
(2.17)
~∆
By definition of the polarizability α(ω) (Eq. (2.8)) the scattering of photons and the
dipole force are interconnected and are strongest for near resonant light. However,
photon scattering from the dipole trap lasers should be avoided since it leads to
heating of the atoms and shortens the coherence and the lifetime of atomic states.
In order to minimize the scattering rate (Eq. (2.17)) while maintaining strong dipole
forces, far off resonant trapping laser fields (|∆| ≫ Γ) with sufficiently high intensities have to be used.
2.1 Lorentz model
2.1.3
35
Multilevel atoms
For large detunings of the trapping laser fields, the multilevel structure of real atoms
and the multitude of transitions to various energy eigenstates have to be taken
into account. Therefore, the application of the classical Lorentz model (Eq. (2.1))
encounters its limits. However, in a semiclassical picture, with a multilevel atom
with n dipole-allowed transitions from the ground state to excited states, each
transition can be approximated as an independent harmonic oscillator. The general
solution can be constructed as a sum of the contribution of all individual oscillators
weighted by the oscillator strength fj of each transition, yielding
α(ω) =
n
X
fj αj (ω).
(2.18)
j=1
The oscillator strength can be calculated by modeling the atomic wave function or
determined from the experimentally obtained data of transition line widths [49].
For the cesium D-lines, the latter method is more precise because the excited 6P
states decay only to the 6S1/2 ground state. The relation between the linewidth
(decay rate) Γj and the oscillator strength fj reads
f j = Γj
2πε0 me c3 ge
,
e2 ωj2 gg
(2.19)
where ωj = (Eej − Eg )/~ denotes the transition frequency of the j-transition and
ge /gg = (2J ′ + 1)/(2J + 1) is the ratio between the degeneracies of the excited and
ground states. Hence, the real and complex parts of the polarizability become
Re[α(ω)] = 2πε0 c3
X 2J ′ + 1
j
Im[α(ω)] = 2πε0 c3
2J + 1 (ωj2 − ω 2 )2 + Γ2j (ω 3 /ωj2 )2
X 2J ′ + 1
j
(ωj2 − ω 2 )Γj /ωj2
Γ2j ω 3 /ωj4
2J + 1 (ωj2 − ω 2 )2 + Γ2j (ω 3 /ωj2 )2
,
(2.20)
.
(2.21)
The resulting scattering rate as well as the dipole potential of a cesium atom in the
ground state can now be obtained analogously to Eqns. (2.8) and (2.14). Note that
the calculation of potentials of excited states requires a full quantum mechanical
description of the light-matter interaction (see next section).
36
E
|ei
~ω0
U2
|gi
U1
(a)
(b)
(c)
Figure 2.2: AC-Stark shift of a two-level atom in a red-detuned and bluedetuned laser beam, respectively. (a) Unperturbed level diagram of the ground
state |gi and excited state |ei. The red and blue arrows indicate the respective
detuning. (b) Shift of the ground and excited state in the presence of red-detuned
light. Dark regions indicate high intensities. Note that an excited atom is repelled
from the trap region. (c) For blue detuning the shifts are inverted.
2.2
AC-Stark shift
One of the main motivations of laser cooling and trapping is the storage of atoms
over times sufficiently long enough to efficiently store and retrieve quantum information (qubits) encoded in the atomic states using (near) resonant light. Due to
the weak coupling of neutral atoms to their environment, coherent manipulation
of atomic states can be robust against external perturbations [5]. Therefore, the
feasibility of storing atoms, almost without loss of the atomic coherence, makes optically trapped neutral atoms prime candidates for the implementation of quantum
memories and quantum repeaters in the context of quantum information processing
and transmission [23,27]. For this purpose, far off-resonant detuned lasers are used
which ensure low scattering rates and thus provide a long coherence time. However,
the interaction of the trapping light field inherently perturbs the atomic states and
can influence the interaction with resonant light.
The presence of an intense red-detuned light field lowers the ground state energy of
the atom (see Fig. 2.2). The lowest potential energy of the atom, i.e., the location
where the atom can be trapped, is associated with the largest negative shift, and
is found at the local intensity maximum. In general, the light shift of the excited
state is different from that of the ground state. Thus, in strongly confining traps
the differential light shift (ac-Stark shift), of the connected pair of states leads to
a strong position dependent shift accompanied with inhomogeneous broadening of
the atomic transition. Moreover, for resonantly driven atoms the trapping potential
can be even repulsive while the atom is excited, leading to dipole force fluctuation
and heating.
2.2 AC-Stark shift
37
For blue-detuned light the ac-Stark shift leads to an increase of the atomic potential
energy which, for special beam geometries (e.g., hollow beams, optical lattices etc.),
enables trapping of atoms in the region of minimum intensity [40, 58]. In this case,
the differential light shift can be locally minimized. However, due to diffraction of
such strongly confined laser beams the potential minimum is never completely field
free. Thus, confinement in blue-detuned traps is less efficient and more prone to
inhomogeneous broadening.
The straightforward approach to avoid the perturbation of the trapping fields is to
switch off the trapping light and to perform the measurement on the free-falling
atoms [29]. However, a drawback of this method is the limited time-of-flight and the
short recapture range of optical traps. Another approach employs so-called magic
wavelengths in red-detuned traps in order to eliminate the differential ac-Stark
shift [60,61]. The principle idea is based on a clever choice of the wavelength of the
trapping light for which an atom experiences shifts of equal sign and magnitude of
the excited state and of the ground state.
In Ref. [65] it has been shown theoretically that the differential light shifts for
cesium atoms, subjected to the combined red-detuned and blue-detuned trapping
fields in a nanofiber trap, can be eliminated analogously.
However, the theoretical approach to the description of the ac-Stark shift in a
nanofiber trap is incomplete. The model in Ref. [65] is based on the formalism
derived for the static Stark shift in [66], which separates the interaction into a
monopolar (scalar) and a quadrupolar (rank 2 tensor) symmetry, whereas the dipolar (axial vector) part is neglected. Meaning that, the extension of this model to
the case of a dynamic interaction is only valid for linearly polarized light fields.
Since the polarization of the evanescent field of a strongly guided mode HE11 varies
in space and time [32, 34], the treatment of the ac-Stark shift in the nanofiber trap
thus requires a more general model. An appropriate description of the ac-Stark
shifts induced by arbitrary polarized light fields can be found in Ref. [62] and [67]
and is closely followed here.
In this work we analyze a generalized model describing the ac-Stark shift in opticallytrapped cesium atoms. This allows us to find the perturbed eigenenergies and
eigenstates in the presence of both trapping light fields in a nanofiber trap.
Based on these results, we estimate the influence of the ac-Stark effect on the transition amplitude between the ground and excited states in order to determine the
coupling efficiency as well as the absorption spectra of the trapped atoms interacting with a resonant probe light. We show that for certain cases the trapped atoms
exhibit shifts of the energy levels with a negligible mixing of the atomic states.
Furthermore, we demonstrate that the experimentally obtained absorption spectra
for trapped cesium atoms are in good agreement with the numerical calculation of
the light shifts.
38
Theory
The general quantum mechanical description of atoms subjected to a monochromatic laser field makes use of the “dressed states” approach [63], in which both the
atomic states as well as the light field are quantized. For light waves that can be
described by a single radiation mode with large photon numbers, the field equations
take the equivalent form as for a classical electromagnetic wave.
In this work we will thus consider the interaction of an atom and a classical optical
field
1
(2.22)
E(r, t) = Eee−iωt+ik·r + c.c.,
2
where c.c. designates the complex conjugate of the preceding term, E denotes the
electric field amplitude, ω is the angular light frequency, k is the wave vector, and e
is the complex polarization vector. In the dipole approximation the coupling term
of the Hamiltonian in the interaction picture reads
1
VE ≡ −E · d = − Ee · de−iωt + H.c.
2
= V− e−iωt + V+ eiωt ,
(2.23)
where
d=
N
X
eri
(2.24)
i
is the electric dipole moment operator and ri represents the position vector of the
ith electron and H.c. represents the Hermitian conjugate. The total Hamiltonian
of the system “atom + field” is thus given by
H = H0 + VE
= (Helect. + Vhfi ) + V− e−iωt + V+ eiωt ,
(2.25)
where H0 = Helect. + Vhfi denotes the atomic Hamiltonian accounting for the electronic and hyperfine structure.
For atoms with a finite nuclear and electronic angular momentum I and J, respectively, the I · J coupling leads to the energy eigenstates represented by the hyperfine
states (hfs) |Ψi = |n, F, MF i, which are angular momentum eigenstates of the operator F = J + I. Here, the quantum number F is referred to the total angular
momentum, MF is the projection of F onto the quantization axis, and n denotes
all remaining quantum numbers {n(ls)JI} characterizing the (fine) structure of
the atoms. Compared to the level splitting arising due to the hyperfine interaction (Vhfi ), the effect of the ac-Stark shift is assumed to be small. Hence, we may
treat the energy correction of the atomic states in the presence of a weak variable
electromagnetic field in terms of time-dependent perturbation theory.
2.2 AC-Stark shift
39
For periodically varying fields, this treatment gets even simpler if we apply Floquet’s theorem [62, 64]. By employing the basis of quasienergy states of the system
composed of an atom and a field, we can use the result for the Stark shift in a static
electric field
X | hΨ0 | Ee · d | Ψi i |2
∆E0 =
,
(2.26)
E0 − Ei
i6=0
however, instead of using the unperturbed energies Ei of the intermediate states
of the atom we have to employ the energies Ei + ~ω and Ei − ~ω of the complete
system. These two energies are referred to the absorption and emission of a single
photon in the transition from the initial state |Ψ0 i = |n, F, MF i to an intermediate
state |Ψi i = |n′′ , F ′′ , MF′′ i over which the summation in Eq. (2.26) is performed.
Note that higher orders of the perturbation theory correspond to the absorption
and the emission of a larger number of photons.
Hence, by applying time-independent second-order perturbation theory with an
effective electric field amplitude (depending on the polarization of the field (cf.
Eq. (2.22)), the following expression for the energy correction (ac-Stark shift) of
the atomic state |Ψ0 i is obtained
1 X hΨ0 | e∗ · d | Ψi i hΨi | e · d | Ψ0 i
∆Eac = E 2
4
E0 − Ei + ~ω
i
1 X hΨ0 | e · d | Ψi i hΨi | e∗ · d | Ψ0 i
,
+ E2
4
E0 − Ei − ~ω
(2.27)
i
where we sum over the non-degenerate sub-space and make use of the abbreviation
i = {n′′ , J ′′ , F ′′ , MF′′ }.
By introducing the operator
1
VEE (E, ω) = E 2 {(e∗ · d)RE0 (ω)(e · d) + (e · d)RE0 (−ω)(e∗ · d)},
4
with
RE0 (±ω) = (E0 − Ĥ0 ± ~ω)−1 =
X
i
|Ψi i hΨi |
,
E0 − Ei ± ~ω
(2.28)
(2.29)
being the resolvent operator (Ĥ0 is the unperturbed atomic Hamiltonian), the energy shift in Eq. (2.27) can be recast as an expectation value
∆Eac = hΨ0 |VEE (E, ω)|Ψ0 i ,
(2.30)
while the contributions due to mixing of states are accounted by the off-diagonal
matrix elements hΨ0 |VEE (E, ω)|Ψ′0 i.
40
In general VEE (E, ω) is not diagonal in the hfs basis, thus in order to find the
new eigenstates and eigenenergies one has to diagonalize the atomic Hamiltonian
including the ac-Stark interaction. However, we assume relatively small light shifts
in a dipole trap compared to the large hyperfine splitting of the relevant |n, F, M i
states (∆Ehfi > 150 MHz for the 6P3/2 states).
Under the assumptions of small light shifts VEE can be regarded as diagonal in
|n, F i. In this case, only the matrix elements between the states |n, F, M i and
|n, F, M ′ i need to be considered for the further calculation (i.e., replace F ′ with F
in the following calculations).
Furthermore, the order of the operators in Eq. (2.28) may be changed and rewritten
in a spherical tensor form [67]
h
i
X
1
{e∗ ⊗ e}K · {d ⊗ RE0 (ω)d}K + (−1)K {d ⊗ RE0 (−ω)d}K ,
VEE (E, ω) = E 2
4
K=0,1,2
(2.31)
leading to a decomposition of VEE into a scalar (K = 0), vector (K = 1), and
tensor (K = 2) part.
In the following we will abbreviate the notation for the matrix elements of VEE by
TM,M ′ = hn, F, MF |VEE (E, ω)|n, F ′ , MF′ i .
(2.32)
By employing the Wigner-Eckart theorem we find an explicit expression for each
matrix element
X
1
(K)
F ′ ,M ′
TM,M ′ = E 2
{e ⊗ e∗ }K,µ CF,M,K,µ
αnF (ω)
4 K=0,1,2
µ=−K...K
1 2 X
F
K+µ ∗
F −MF
= E
(−1)
{e ⊗ e}K,−µ (−1)
−M
4 K=0,1,2
µ=−K...K
K F′
(K)
αnF (ω),
µ M′
(2.33)
(K)
where αnF (ω) is the reduced atomic polarizability
(K)
αnF (ω) = n, F {d ⊗ RE0 (ω)d}K + (−1)K {d ⊗ RE0 (−ω)d}K n, F ′ . (2.34)
(K)
The expression for αnF can be conveniently recast and evaluated via
√
√
√
F K F′
(K)
I−J+F
′
αnF (ω) = (−1)
2K + 1 2F + 1 2F + 1
J I J
X J K J
′′ ′′
′′ ′′
×
′′ 1 hnJkdkn J i hn J kdknJi
1
J
n′′ J ′′
1
(−1)K
×
+
,
En,J − En′′ ,J ′′ + ~ω En,J − En′′ ,J ′′ − ~ω
(2.35)
2.2 AC-Stark shift
41
where the reduced electric dipole matrix elements hnJkdkn′′ J ′′ i can be obtained
via the oscillator strengths fnJ,n′′ J ′′ for the |nJi ↔ |n′′ J ′′ i transitions (cf. [50])
fnJ,n′′ J ′′ =
2me EnJ − En′′ J ′′
| hnJkdkn′′ , J ′′ i |2 .
3~2 e2
2J + 1
(2.36)
Here, we employed the notations CjJ,M
for Clebsch-Gordan (CG) coefficients,
1 ,m1 ,j2 ,m2
n
o
j1 j2 J
j1 j2 j3
and m1 m2 M , j4 j5 j6 for the Wigner 3-j and 6-j symbols, respectively.
P
K,µ
ei (e∗ )j accounts for the spatial oriThe compound tensor {e ⊗ e∗ }K,µ = i,j C1,i,1,j
entation and polarization of the light field. The parametrization of the polarization
in Cartesian coordinates by the angle θ,
e = ex sin θ + iey cos θ,
(2.37)
which is related to the degree of linear polarization l = cos 2θ, and to the ellipticity
A = sin 2θ, reduces the tensor {e ⊗ e∗ }K,µ to
1
1
{e∗ ⊗ e}0,0 = − √ (e∗ · e) = − √
3
3
1
A
{e∗ ⊗ e}1,µ = − √ (e∗ × e)µ = − √ δµ,0
2
2
1
l
{e∗ ⊗ e}2,µ = − √ δµ,0 + √ δµ,±2 .
6
2
(2.38)
Note that the quantization axis (z-axis) is chosen as the propagation direction of
the light.
The 3-j symbol in Eq. (2.33) indicates the coupling of the hfs states with different
MF quantum numbers through the ac-Stark interaction and makes the operator
VEE in general non-diagonal in this sub-space. Nevertheless, for linearly and circularly polarized light, the correct choice of the quantization axis makes TM,M ′
diagonal.
If the field is oriented in such a way that an axis of rotational symmetry of the
system exists, i.e., z-axis along propagation direction for circular polarization, and
z-axis along the electric field E for linear polarization, MF is a constant of motion. Thus, the off-diagonal elements of TM,M ′ vanish and |F, MF i become also
eigenstates of the Stark operator.
In the case of linear polarization, we find similar expressions for the diagonal energy
correction as for the case of static electric fields, involving a scalar (αsnF ) and a
tensor (αTnF ) term [66]
3MF2 − F (F + 1)
1 2 s
T
(2.39)
∆Eac = TM,M = − E αnF (ω) − αnF (ω)
.
4
2F (2F − 1)
42
For the case of circular polarization (i.e., A = ±1), the additional vector term
(αanF MF ) shifts the energy levels in analogy to the Zeeman shift, yielding following
expression for the ac-Stark shift
3MF2 − F (F + 1)
1 2 s
MF
a
T
− αnF (ω)
,
∆Eac = TM,M = − E αnF (ω) + AαnF (ω)
4
2F
2F (2F − 1)
(2.40)
with
αsnF = p
αanF
αTnF
1
(0)
αnF ,
3(2F + 1)
s
2F
(1)
=−
αnF ,
(F + 1)(2F + 1)
s
2F (2F − 1)
(2)
α .
=−
3(F + 1)(2F + 1)(2F + 3) nF
(2.41)
The expression for the ac-Stark shift (Eq. (2.40)) holds only if F and MF are
good quantum numbers, which in practise can be assured by applying a quantizing
magnetic field along the propagation direction of the light.
For the general case of elliptical polarization (i.e., |A| < 1), MF is not a constant
of motion. Thus, the off-diagonal elements of the following form have to be taken
into account
TM,M ±2 =
3E 2 T
[(F ± M + 1)(F ± M + 2)(F ∓ M )(F ∓ M − 1)]1/2
lαnF (ω)
.
16
F (2F − 1)
(2.42)
With the final form of the reduced polarizabilities (Eq. (2.35)), it can be easily
proven that for J = 0, 1/2 states (as the ground states of alkali-atoms such as Cs
(2)
and Rb with J = 1/2), the tensor polarizability αnF vanishes, and for linearly
polarized light the ac-Stark shift is only determined by the scalar component in
Eq. (2.39), which is independent of the orientation of the quantization axis and
of the MF quantum number. The explanation for this behavior is based on a
symmetry argument, i.e., the fact that the quadratic Stark shift for electrons with
an angular momentum of Jz = 0, ±1/2 is independent of the spatial orientation of
the field. Hence, the corresponding hfs |F, Fz i which are linear superpositions of
these electronic states times nuclear states are all shifted equally.
For states with J > 1/2, however, the orientation of the electric field with respect
to the quantization axis is of relevance to the tensor shift. This leads to optical coupling between MF′ = MF ± 2 states and lifts the degeneracy of the new
eigenstates.
2.3 Light shifts and potentials in a two-color dipole trap
2.3
43
Light shifts and potentials in a two-color dipole
trap
In the following we consider the energy shifts, i.e., potentials of the ground and
excited states of cesium with J = 1/2 and J = 3/2 respectively, subjected to two
light fields of different wavelengths and different polarizations. This analysis is
devoted to the estimation of the light shifts of the |6S1/2 i and |6P3/2 i states in
order to examine the perturbation of the D2-transition of optically trapped cesium
atoms in a two-color evanescent light field around a nanofiber.
For this dipole trap, the intense evanescent fields around the nanofiber lower the
ground state energy of the atoms in the vicinity of the fiber surface, and thus induce
the dipole trapping potential. The red-detuned light field as well as the van der
Waals force attracts the atoms towards the nanofiber while the blue-detuned light
field forms a repulsive potential and repels the atoms from the fiber surface [15,31].
Due to the different radial decay lengths of the far red- and blue-detuned evanescent
fields of the fundamental HE11 mode one can thus create a radial potential minimum
at a few hundred nanometers from the nanofiber surface by properly choosing the
respective powers (see Fig. 2.3 (a)).
Confinement along the fiber axis is achieved by launching an additional, counterpropagating red-detuned laser beam through the fiber, thereby realizing a reddetuned standing wave, see Fig. 2.3 (b), whereas the blue-detuned running wave
leads to a translation invariant repulsive field along the nanofiber. Azimuthal confinement of the atoms stems from the azimuthal dependence of the evanescent field
intensity of the quasi-linearly polarized HE11 mode (see Fig. 2.3 (c)). In order to
maximize the azimuthal confinement we use orthogonal linear polarizations, i.e., the
azimuthal intensity maximum of the red-detuned field coincides with the azimuthal
intensity minimum of the blue detuned field.
Figure 2.3 (d) shows the two resulting 1d arrays of trapping minima on both sides
of the fiber, visualized by the equipotential surfaces 40 µK and 125 µK above the
trapping minimum.
In this configuration, the potential minima can be found at the anti-nodes of the
red-detuned standing wave in the plane of maximum intensity which is parallel to
the polarization direction of the red-detuned light field. At these points, the bluedetuned field is polarized tangential to the fiber surface, making both trapping
fields purely linearly polarized in the vicinity of the trapping minimum [31]: The
(longitudinal) z-components of the red-detuned evanescent fields interfere destructively at the anti-nodes due to their phase shift of π/2 with respect to the transverse
components (see Sect. 1.1). In addition, the z-component of the blue-detuned field
vanishes because its polarization vector is oriented parallel to the fiber surface (see
Sect. 1.3.2).
44
4
0.2
-4
-0.2
-8
-0.4
-12
-0.6
-250
- 16
-0.8
Potential (MHz)
0.4
z (nm)
500
8
250
0
-500
500 250
0
0 250 500
Distance (nm )
> 0.0
- 0.1
y
- 0.2
- 0.3
y
x
z
< - 0.4
x
Figure 2.3: (a, black line) Light-induced potential plus the van der Waals potential as a function of distance from the surface of a 500-nm diameter nanofiber
for a ground state cesium atom (S1/2 ) formed by a red-detuned (wavelength 1064
nm, red line) and blue-detuned (wavelength 780 nm, blue line) two-color evanescent light field. We assumed a power of Pred = 2 × 2.2 mW (standing wave) and
Pblue = 25 mW and orthogonal linear polarization of the red-detuned and bluedetuned fields, indicated by the arrows in Fig.(c). The green dashed line indicates
the v.d.W potential solely. (b) Contour plots of the same potential as in (a). The
red-detuned standing wave ensures axial confinement. (c) Azimuthal plot of the
same potential as in (a) and (b). The planes of the plots in (a)–(c) are chosen such
that they include the trapping minima in all three cases. (d) Contour plot of the
resulting array of trapping sites on both sides of the fiber showing equipotential
surfaces 40 µK and the 125 µK above the trapping minimum.
45
The ground state potential for cesium atoms, induced by the linearly polarized
evanescent fields in the mentioned configuration, as a function of cylindrical coordinates (r, φ, z) reads
1
2
2
U (r, φ, z) = − 4αsg (ωred )Ered
(r, φ) cos 2 (βred z) + αsg (ωblue )Eblue
(r, φ + π/2) ,
4
(2.43)
where the radial and azimuthal dependency of the respective electric fields E(r, φ) is
given by Eq. (1.52) in Sect. 1.3.2, βred = neff kred is the propagation constant of Ered
and thus the lattice constant of the standing wave potential, neff is the effective
refractive index, kred is the wave number of the red-detuned light field [34], and
αsg (ω) is the scalar polarizability of the ground state. Note that admixtures of
circularly polarized light lift the degeneracy of the Zeeman sub-states, i.e., the
ground state potential becomes MF dependent. Nevertheless, at the vicinity of
the trap center, where the polarizations of the fields are essentially linear, equation
(2.43) remains valid. The strongest deviation from linear polarization for the reddetuned field can be found for small displacements in the axial direction. The
estimated thermal motion of atoms in our trap results only in a displacement of
about 12 nm from the trap center (see Sect. 3.4) where the ellipticity is negligibly
small (i.e., A ≈ 0.002).
We also note that the contribution of the v.d.W. potential to the trapping potential
at radial distances beyond 100 nm from the fiber surface can be completely neglected
(see Fig. 2.3 (a)).
Numerical evaluation
For the calculation of the light-shifts of the ground and excited state manifold of
cesium |6S1/2 , F = 4, M i, and |6P3/2 , F = 5, M i respectively, we applied the formalism discussed in the previous section. We diagonalized the Hamiltonian of this
system which includes the the Stark interaction of each light field. For the calculation of the ac-Stark shifts of the |6P3/2 , F = 5i manifold, we considered the dipole
allowed coupling to the states listed in Fig. 2.4. The corresponding polarizabilities
have been derived from Eq. 2.35 and the oscillator strengths given by [50].
Figures 2.3(a) and 2.5 show the resulting energy shifts of the respective states of
atomic cesium in the vicinity of a 500-nm diameter fiber. Here, we assumed the
powers and wavelengths (780 nm with 25 mW, and 1064 nm with 2 × 2.2 mW) that
are used in the experimental part of this work. These numerical results have been
obtained from the eigenvalues of the summed tensors of both fields, i.e., the solutions
blue + T red − ∆E δ
of the eigenvalue problem det kTM,M
′
ac M,M ′ k = 0 in Eq. (2.33) and
M,M ′
Eq. (2.43) evaluated using Mathematica.
The resulting light-shift of the ground state at the trap center (r = 230 nm, φ = 0,
z = 0) yields −8 MHz and the shift and splitting of the excited state manifold (see
Fig. 2.6) yield −11 to +9 MHz at the trapping minimum, depending on the new
|F ′ , m′ i sub-state.
46
Figure 2.4: Level diagram and transition wavelengths of atomic cesium. The
6P3/2 state couples additionally to the ground state also to a multitude of energetically higher lying states, which have to be taken into account for the calculation
of its light shift.
∆E [MHz]
30
20
10
r [nm]
100
200
300
400
500
- 10
- 20
- 30
- 40
Figure 2.5: Optical potentials for the |6P3/2 , F = 5, M i sub-state manifold including the v.d.W. potential as a function of the radial distance from the fiber
surface. Note that the degeneracy is lifted by the presence of two mutually orthogonally polarized light fields.
47
∆EJ=3/2,F =5,M [MHz]
10
5
.
-5
-5
-10
.
....
.
.
.
0
.
5
.
f′
M
-15
∆EJ=1/2,F =4,M [MHz]
-4
0
4
M
-5
-10
Figure 2.6:
AC-Stark shift of the |6S1/2 , F = 4i ground state and the
|6P3/2 , F = 5i excited state manifold of atomic cesium at the trap center, located
230 nm above the surface of the fiber. The quantization axis is aligned along
the polarization direction of the leading field in the trap (E red ). The red squares
indicate the light shifts predicted by the reduced model, i.e., when neglecting the
off-diagonal coupling (see text).
Results and Discussion
From the definition of the reduced polarizability in Eq. (2.35) and symmetry arguments, it can be shown that for linearly polarized light the J = 1/2 ground
state exhibits only a non-vanishing scalar contribution (αsg ) to the ac-Stark shift,
which maintains the |n, J = 1/2, F, MF i states as eigenstates including the degeneracy of the MF sub-levels. Note that elliptically polarized light, as well as a
non-vanishing (longitudinal) Ez -component of the HE11 mode, introduce a “fictitious magnetic field” due to the vector polarizability (cf. Eq. (2.33)). This breaks
the degeneracy through a Zeeman-like shift and creates coherences among the MF
sub-states. However, the field configuration in our setup features essentially linear
polarizations at the trapping minima. Hence, the electric dipole transitions between
J = 1/2 ↔ J ′ = 1/2 states (e.g., D1-line |6S1/2 i ↔ |6P1/2 i) are expected not to be
significantly influenced by the presence of two linearly polarized trapping fields.
In contrast to the J = 1/2 states, the non-vanishing tensor component (αTe ) makes
the potential of the J = 3/2 excited states in a optical dipole trap strongly dependent on the F and MF quantum numbers including a change in sign (see Fig. 2.5).
For a single linearly polarized light field (polarized along quantization axis), the
ac-Stark shift scales quadratically in MF (cf. Eq. (2.40)), leaving a two-fold de-
48
generacy corresponding to the pairs of MF numbers with the same magnitude (but
opposite sign).
However, the presence of a second orthogonally polarized light field breaks the
rotational symmetry of the system, such that no orientation of the quantization
axis can be found for which the excited states |J = 3/2, F, MF i are eigenstates. In
this case the non-vanishing off-diagonal coupling (TM,M ±2 6= 0 in Eq. 2.33) gives
rise to mixing between the degenerate |MF i sub-states. The new energy eigenstates
thus become superpositions of these sub-states
X
|n, F, mi =
cMF ,m′ |n, F, MF i ,
(2.44)
MF
where the mixing coefficients cMF ,m of the eigenstates in the |MF i basis can be obtained by diagonalization of the Hamiltonian in the presence of both light fields in
Eq. (2.33). The mixing of the excited states may result in simultaneous coupling to
various hyperfine states via the J = 1/2 ↔ J ′ = 3/2 transitions, which may influence the resonant atom-light interaction in our system in addition to inhomogenous
broadening.
Nevertheless, for MF states defined with respect to the quantization axis chosen
along the polarization vector of the dominating field (red-detuned field) the effect of
hfs mixing can be neglected. For the trap parameters considered in the experimental
part, we find only a small deviation of the ac-Stark shift of the |6P3/2 , F = 5, MF i
states of at most 1 MHz out of 9 MHz less shift for the low MF numbers (and
0.03 MHz deviations for MF = ±F states) when neglecting the off-diagonal matrix
elements of the Stark operator (TM,M ±2 → 0) (see Fig. 2.6).
In addition, a weak bias magnetic field applied parallel to the red-detuned field
could be used in order to insure that MF is still a good quantum number. Provided
that the off-diagonal coupling of the ac-Stark interaction induced by blue-detuned
field is weaker than the splitting of the Zeeman levels, mixing of the MF states can
be completely neglected [67].
For a better understanding of the experimental results regarding the shifts and
broadening of the absorption spectra of trapped cesium atoms under the influence
of the two-color dipole fields we applied the formalism discussed above. The experimentally observed spectra for the D2 transitions in a trapped atomic ensemble with
evenly populated MF states are in good agreement with the theoretical predictions
of the ac-Stark effect even when assuming negligible mixing of hfs (see Sect. 4.1).
Chapter 3
Trapping of neutral atoms
Introduction
In view of the excellent coherence properties of neutral atoms, which are essential
for many applications in quantum optics and quantum information technology, they
are one of the most promising candidates as carrier and storage units of quantum
information [6].
However, in general the control of neutral atoms is technically challenging, due
to the relatively weak interaction of the induced electric dipoles with electromagnetic fields. For this reason alkali atoms, i.e., sodium, rubidium, cesium, etc. are
predominantly employed, because of their strong interfacing to light via the Dtransitions. These strong electric dipole transitions provide almost perfectly closed
sub-transitions and are accessible with tunable lasers, making alkali atoms suitable
for laser cooling and for optical trapping (see Sect. 3.1).
Light-induced dipole forces act in general on any polarizable particle in a strong
gradient electro-magnetic field and have been proposed for trapping of particles
since the first realization of a laser in the 1960s [70, 71]. The first applications
of these forces have been found in, e.g., biology as optical tweezers. However,
light induced dipole forces exerted on neutral atoms are very weak, thus only very
cold atoms can be considered for optical trapping. In this context, T. Hänsch,
A. Schawlow [72], D. Wineland and H. Dehmelt [73] independently proposed already
in 1975 the cooling of atomic gases with near resonant laser light.
The first three-dimensional laser-cooling scheme for atoms was realized by S. Chu
[75] in 1985, permitting the first experimental demonstration of optical trapping
of neutral atoms [76]. The development of laser-cooling in combination with the
magneto-optical trap (MOT) [77] led to the utilization of dipole traps for atoms in
many varieties. Among focused laser beams, evanescent waves on prisms [78–80] or
50
planar wave guides [84] have been readily applied, from trapping of ensembles of
neutral atoms to forming of an all-optical Bose-Einstein condensate (BEC) [80].
Recently, optical nanofibers with diameters smaller than the wavelength of the
guided light have attracted considerable interest in the field of quantum optics
due to their high potential as light-matter interfaces. Because of the strong light
confinement and thus high intense evanescent fields featured by a nanofiber for
almost unlimited lengths, these fibers are ideal tools for both trapping of atoms
and interfacing them to light.
In this context, various theoretical works, e.g., by Dowling and Gea-Banacloche [30],
and Balykin et al. [31] followed this idea, and have proposed to realize a two-color
optical dipole trap which makes use of a red-detuned and blue-detuned evanescent
light field surrounding the optical nanofiber.
In this work, I demonstrate the first realization of a trap for neutral cesium atoms
in the evanescent field of a nanofiber via light induced dipole forces in combination
with efficient interfacing of the atoms with the fiber guided light. In the coming
chapter, I introduce prerequisites of atom trapping and present our experimental
setup. Furthermore, I report on the experimental characterization of our nanofiberbased trap.
3.1
Magneto-optical trap (MOT)
The magneto-optical trap (MOT) is nowadays used as a standard source of cold
atoms. It became a versatile tool for many atomic physics applications: It allows
one to capture and cool atoms from a dilute gas at room-temperature down to
about hundred microkelvin [5], to store them for a certain time or to transfer atoms
directly into other traps.
For this work, a three-dimensional MOT is employed in order to accumulate a
dense cloud of cold cesium atoms around the nanofiber. By carefully adapting the
parameters of the operation of the MOT a dense sample of the cold atoms can be
efficiently loaded into the trapping sites surrounding the nanofiber.
The working principle of a MOT is based on cooling and trapping using spatially
modulated scattering forces induced by near resonant laser light. Here, the laser
cooling mechanism relies on velocity-dependent cooling forces which dissipate the
kinetic energy of the atoms, and on position-dependent restoring forces which lead
to spatial confinement of the atoms.
3.1 MOT
51
Figure 3.1: Doppler cooling. (a) The Doppler effect leads to preferential photon
scattering from the counter-propagating beam, whereas the direction of spontaneous emission is random. (b) Light forces versus velocity for a detuning of
∆ = −Γ. The functional form of the scattering rate of the atoms leads to
a Lorentzian-like force dependency on velocity (dashed lines) with maxima at
v = Γ/k. The cooling force results from the combination of both forces (red solid
line).
3.1.1
Doppler cooling/Velocity dependent forces
Doppler cooling can be well explained by the picture of an atom moving parallel
to two counter-propagating laser beams which are slightly red-detuned (∆ < 0)
with respect to the atomic resonance. The Doppler shift of the resonance frequency
of a moving atom compensates for the detuning and leads to preferential photon
scattering of the counter-propagating laser beam. The absorption of one photon by
the atom implies the change of the atomic momentum by ~k, where k is the wave
number of the incident light. In the subsequent spontaneous decay, the reemission
direction of the photon is arbitrary. Hence, for many such scattering events the
averaged recoil momentum per photon hpi = ~k points into the propagation direction of the laser beam, leading to a friction-like slowing down of the atoms (see
Fig. 3.1). The cooling force F (v) results from the combined light forces induced by
the counter-propagating laser beams and is determined by the scattering rate Rs ,
the average recoil momentum ~k, and the doppler shift kv
where
F (v) = ~k[Rs (I, ∆ − kv) − Rs (I, ∆ + kv)],
(3.1)
"
#−1
ΓI
I
2∆ 2
Rs (I, ∆) =
1+
+
,
2 I0
I0
Γ
(3.2)
(see Sect. 2.1). The Lorentzian dependence of the scattering rate on the detuning
(Eq .(3.2)) leads to a Lorentzian velocity dependence of the cooling force with a
maximum (minimum) at v = (−)∆/k for the respective beam. For atoms within
this velocity range, the cooling force is linear in v (F = αv), leading to a viscous
drag.
52
Figure 3.2: (a) One dimensional scheme of a magneto optical trap for atoms
with ground and excited states J = 0 and J ′ = 1, respectively. A magnetic
field Bz = b · z leads to a position dependent energy shift of the Zeeman sublevels (denoted by mJ ). Regarding the selection rules (∆m = ±1) for the σ + , σ −
polarized laser beams (red curly arrows), the laser beam which pushes the atom
towards the center z = 0 is preferentially absorbed. (b) Six circularly polarized
beams crossing at the center of the trap form the 3D-molasses. The linear magnetic
field gradient is created using two coils in anti-Helmholtz configuration (circulating
arrows). The direction of the magnetic field is indicated by grey lines.
Optical molasses
The straightforward extension of the Doppler cooling scheme to three dimensions
is realized by three mutually orthogonal pairs of counter-propagating laser beams
crossing at the trap center. In this intersection region the laser beams provide
viscous friction and form an optical molasses for atoms moving in any direction.
However, the arising viscous friction can not cool the atoms to absolute zero temperature. The randomness of the recoil due to the spontaneous emission leads to
a fluctuation of the momentum around hpi = 0, equivalent to a diffusive drift or a
random walk in phase space. Thus, the atoms undergo heating and cooling at the
same time. In this simple model of laser-cooling, the theoretical minimum temperature that can be reached is determined by the equilibrium of cooling and heating,
which is referred to the Doppler-limit or the “Doppler temperature” defined as
TD = ~Γ/2kB ,
where kB is the Boltzmann constant.
(3.3)
3.1 MOT
53
Position dependent force
The principle of the restoring forces in a MOT is best illustrated by a reduced onedimensional model (see Fig. 3.2a). An atom subjected to a magnetic field whose
magnitude is radially increasing from zero at the trap center, experiences a lift of the
degeneracy of the excited state Zeeman multiplicity and a position dependent shift.
Due to the selection rules (∆m = ±1) for the σ + and σ − polarized laser beams
respectively, an atom displaced from the center gets shifted into resonance with the
beam that pushes the atom back towards the center. This extension features the
great advantage that atoms can be cooled and trapped in a small volume, which
results in a high density of the cold atom sample, such that atoms can efficiently
be loaded into a dipole trap.
A more comprehensive theory accounting for the multilevel structure of real atoms
in conjunction with spatially varying light polarization and magnetic fields predicts cooling even beyond the Doppler limit [5, 86]. In this context so-called
“Sub-Doppler” cooling techniques, e.g., polarization gradient-, Sisiphus-, Raman
sideband-, evaporative cooling etc., have been developed which allow to reach
temperatures even beyond the recoil limit and achieve quantum degeneracy (i.e,
BEC) [5,89]. The D2-cooling transition for cesium yields a Doppler temperature of
125 µK, and the measured MOT temperature is typically 70−100 µK. These values
are lower than the depth of the optical potential (U/kB ∼ 0.4 mK) formed by the
evanescent field of the nanofiber, meaning that the temperature is in principle low
enough to confine the atoms inside the trap.
However, the temperature quantifies only the mean kinetic energy, the corresponding Maxwell-Boltzmann energy distribution in the MOT covers atoms with much
higher energies which escape from the trap. Such an “evaporative” escape of the
atoms from the high energy tail of the Maxwell-Boltzmann distribution leads to a
decrease of the thermal energy of the remaining atomic ensemble.
For interacting atoms, i.e., when atoms collide during the loading, the ensemble can
re-thermalize. In this case the temperature equilibrates at a value for which the
atoms experience an effectively infinitely deep potential well, typically about 1/9-th
of the trap depth U0 for a three dimensional potential well [87]. As a consequence,
the temperature of the captured atoms in the nanofiber trap is expected to be
lower than the temperature in the MOT (see Sect. 3.4). Note that in absence of
interatomic collisions the energy distribution is truncated at E0 = U0 , thus without
re-thermalization the temperature can be much larger than 1/9U0 .
54
F′ = 5
251,0 MHz
′
201,2 MHz F ′ = 4
F
151,2 MHz ′ = 3
F =2
62 P3/2
D2 -line
852.3 nm
cooling transition
repump transition
F =4
62 S1/2
9,193 GHz
F =3
Figure 3.3: Hyperfine level scheme of the D2-line of the caesium atom. The
arrows indicate the cooling and repump transitions.
3.2
Experimental setup
3.2.1
Cooling-Laser system
Efficient laser-cooling of atoms requires the light field formed by a single transversal
radiation mode in each spatial direction, as well as frequency stability of the order
less than the linewidth of the cooling transitions. For cooling in our cesium-MOT
we employ the strongest of the D2-transitions (i.e., 6S1/2 , F = 4 → 6P3/2 , F ′ = 5)
at 852 nm wavelength. The optical setup is depicted in Fig. 3.5. As a laser-light
source we use a tapered amplifier system (TA) which is seeded by a frequency
stabilized external cavity diode laser (ECDL) in Littman configuration (Sacher,
TEC-420). The frequency stabilization is realized by a Doppler-free saturation
spectroscopy [90, 92] and an electronic feed-back loop to the piezo actuator of the
external cavity, yielding a long term stability of about 1 MHz. The amplified
laser beam passes twice through an acousto-optical modulator (double-pass AOM),
which is used for fast and independent control of the laser detuning and power,
necessary for loading the atoms from the MOT into the dipole trap. This cooling
beam is coupled into a single-mode optical fiber and split into a six pathway fiber
cluster (Schäfter+Kirchhoff). Each of these fiber-ports is connected to an output
collimator generating a six beam MOT with a 1/e2 -beam diameter of 13 mm, a
power of up to 30 mW per beam, and polarizations according to the requirements
for position dependent forces (see Fig. 3.2(b)). The hyperfine structure of cesium
with I = 9/2, J = 1/2 and J ′ = 3/2 reveals a relatively small level splitting of
the excited state (see Fig. 3.3), such that the cooling laser which drives the closed
transition F = 4 → F ′ = 5 also weakly excites the F = 4 → F ′ = 4 transition. For
3.2 Experimental setup
55
the latter case, the subsequent decay branches also in the F ′ = 4 → F = 3 channel,
which results in a reduction of the population in F = 4. Since the atoms pumped
into the F = 3 state are too far detuned to be coupled to the cooling beams, a
weak repump laser driving the F = 3 → F ′ = 4 transition is used to repump the
population back to the F = 4 state. The repump laser (ECDL) (Sacher, TEC-100)
is frequency stabilized in the same way as the cooling laser. Its beam is superposed
with the cooling beam using a 90:10 beam splitter (BS) before entering the fiber
cluster, yielding about 120 µW power per beam.
3.2.2
Vacuum setup
The operation of a MOT and a dipole trap critically depends on collisions with background gas molecules; for an adequate trap lifetime of about 1 s ultra high vacuum
(UHV) conditions are mandatory (i.e., pressure < 10−9 mbar). Our setup consists
of a cylindrical stainless steel chamber (∅200 mm × 200 mm) which contains a
moveable holder for the tapered optical fiber (TOF) and a microscope objective
for imaging. Optical access for the MOT beams and the camera is provided by
seven view ports which are anti-reflection (AR) coated for a wavelength of 852 nm,
(reflectivity R < 0.1%). The two ends of the TOF enter and exit the chamber via a
teflon based vacuum feed through [93]. The chamber is attached via a T-connector
(CF150) to a turbo-molecular pump (Pfeiffer TPH180) and an ion-getter pump
(Varian, Starcell 150), providing a pressure below 10−11 mbar without baking the
chamber. An additional UHV gate valve (VAT, 10840) is used to separate the mechanical pumps thus enabling operation of the ion pump solely. The mechanical
pumps including the pre-vacuum pump (Leybold, SC15D) are lubricant-free which
avoids contamination of the vacuum. As a source of cesium, dispensers (SAES
Getters) containing chemically bound cesium are used. They release atomic cesium
when electrically heated at about 600◦ C. The cesium vapor enters the vacuum after
passing an elbow pipe which avoids direct deposition of cesium on the nanofiber.
The presence of cesium raises the pressure to about 8 × 10−10 mbar, indicated by
the current of our ion-getter pump. We note that the pressure indicated by this
gauge with electrodes covered with alkali metals is typically higher than the true
value [94]. When measuring the transmission of a weak laser beam through the
TOF we observe a gradually decrease of transmission due to adsorption of cesium
on the nanofiber. Increasing the laser power to few ten microwatts immediately
restores the initial transmission. Given that much higher powers are used for trapping, possible heating of the nanofiber to few hundreds degree Celsius might cause
desorption of molecules from the fiber surface other than cesium. This would raise
the collision rate with the trapped atoms in the vicinity of the fiber compared to
the rest of the chamber, and would lower the trap lifetime. However, the measured
lifetime of the trap (3.5 s under molasses cooling (cf. Sect. 4.2)) might suggest
unchanged vacuum conditions in the vicinity of the nanofiber.
56
3.2.3
Magnetic fields
The magnetic field gradient in our MOT is realized by the quadrupole magnetic
field created by two coils in anti-Helmholtz configuration along the z-axis ((29) in
Fig. 3.4). The coils have a diameter of 200 mm and carry a current of 8 A providing a magnetic field gradient of 9 Gauss/cm. The magnetic field gradient can
be electronically controlled and switched off within 1 ms (1/e-decay time) using a
field-effect transistor (FET). In order to compensate misalignments due to stray
magnetic fields we use two pairs of Helmholtz coils aligned in the x- and y-direction
with respect to the coordinate system defined in Fig. 3.4. These create a constant
magnetic field, which superimpose on the quadrupole field thus shift controllably
the field zero point in the x, y-plane. Displacement in the z-direction can be compensated by changing the relative currents of the quadrupole coils. The required
currents of the compensation coils have been found after alignment of the MOTbeams and running the MOT with a switching quadrupole field. For an applied
bias field of (2 G, 0, 0) = (Bx , By , Bz ) the MOT-cloud expands isotropically in space
almost without changing its center of gravity. The necessary current for this adjustment is 0.15 A for the coils in the x-direction.
27
30
30
29
40
19
13
UHV pumps
11
17
12
30
29
27
z
x
y
optical table
Figure 3.4: Schematic of the vacuum setup and the electromagnetic coils. The
quadrupole coils are denoted by 29 (horizontal black bars). The compensation
coils are labeled by 30. All other components are listed in Table 3.1.
3.2 Experimental setup
57
41
x
y
z
42
Figure 3.5: Complete schematic of our experimental setup. All components
apart from mirrors are labeled and listed in Table 3.1. The dotted and dashed
arrows indicate the beam path of the red-detuned and blue-detuned laser respectively. Solid lines identify the probe, as well as the cooler and repump laser beams.
58
Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33a,b
34
35
36
37
38
39
40
41
42
Description
cooling laser (Sacher, TEC-420-850)
repump laser (Sacher, TEC-100-850)
probe laser (Sacher, TEC-500-850)
Faraday isolator
λ/2 plate
polarization beam splitter
λ/4 plate
acousto-optical modulator
convex lens
glass substrate
vacuum chamber
fiber mount
fiber positioner
Cs dispensers (SAES-Getters)
avalanche photodiode (APD) (PerkinElmer, C30902E )
photodiode
TOF
fiber coupler
vacuum feed-through for fibers
polarization spectroscopy setup
saturation spectroscopy setup
pre-vacuum pump (Leybold, SC15D)
pre-vacuum valve
microscope objective
camera objective
EMCCD camera (Andor, DU897)
to vertical MOT beam
electrical feed through
magnetic coils
compensation coils
polarization-maintaining optical fiber
beam dump
dichroic mirrors (Laseroptik, custom made)
broad-band mirrors
optical band-pass filters 2×(Thorlabs, BP850)
optical low-pass filter (Thorlabs, LP780)
photodetector
Nd-YAG laser at 1064 nm (Spectra Physics, EXLSR-1064-650)
diode laser at 780 nm (Toptica LD-780-0150)
vibration damper
UHV valve (VAT, 10840)
Fabry-Perot spectrum analyzer
Table 3.1: List of elements labeled in Fig. 3.5.
3.3 Nanofiber trap
59
Figure 3.6: (a) Optical setup of the fiber-based atom trap. The blue-detuned
running wave laser in combination with the red-detuned standing wave laser create
the trapping potential. A resonant laser is used for probing the atoms via the
evanescent field. (b) Fluorescence image of the trapped atomic ensemble.
3.3
Nanofiber trap
Figure 3.6 (a) shows a schematic of our nanofiber trap. We use a laser at a wavelength of 1064 nm which is red-detuned with respect to the D1 (894 nm) and D2
(852 nm) transitions of cesium. Using a beam splitter (BS), two beams are generated and coupled into both ends of the TOF in order to realize the standing wave.
A laser at a wavelength of 780 nm is used for forming the blue-detuned potential
and is superposed with one of the red-detuned laser beams using a phase neutral
dichroic mirror (DM). The probe laser is resonant with the D2 (F = 4 → F ′ = 5)
transition. It is superposed with the blue-detuned laser beam using a second beam
splitter and its power transmitted through the TOF (≈ 1 pW) is measured using an
avalanche photodiode (APD) in combination with an interference filter (IF). Typical powers of the trapping lasers are 2–4 mW for each of the red-detuned beams and
10–35 mW for the blue-detuned beam, depending on the desired trap positions and
depths. In the following, the powers of 2.2 mW for each of the red-detuned beams
and 25 mW for the blue-detuned beam are considered, which, in conjunction with
mutually orthogonal polarizations, yield a trap depth of about 0.4 mK at a distance
of 230 nm from the fiber surface (see Sect. 2.2, Fig. 2.3). The calculated trapping
frequencies for these parameters are 200 kHz, 300 kHz, and 140 kHz in the radial,
axial, and azimuthal direction, respectively. Large detunings of the trapping light
60
fields are chosen in order to ensure a low scattering rate (≈ 10 s−1 ), compatible
with a theoretical trap lifetime due to recoil heating of up to 200 s [31].
The power and detuning of each laser beam in our setup are computer controlled via
the double-pass AOMs. Additionally, the current of the TA can be rapidly switched
off in order to completely extinguish the cooling light. The probe laser has a short
term linewidth of about 1 MHz, which allows us to resolve the natural linewidth
of the F = 4 → F ′ = 5 transition of cesium (5.2 MHz). The state of polarization
of the dipole laser beams at the nanofiber waist can be monitored and aligned
by observing the angular distribution of the Rayleigh scattered light using CCD
cameras (see Sect. 3.3.3). The TOF has been fabricated by flame pulling of a single
mode fiber (Liekki, Passive 6/125) in our computer controlled fiber pulling rig [37].
The nanofiber waist has a homogeneous diameter of 500 nm over its length of 5 mm.
In the tapered sections, the weakly guided LP01 mode of the unstretched fiber is
adiabatically transformed into the strongly guided HE11 mode of the nanofiber
waist and back [11]. This results in a highly efficient coupling of light into and out
of the nanofiber yielding an overall transmission through the TOF of about 99 %
(cf. Sect. 1.4). The lasers are coupled into the ends of the TOF using conventional
fiber couplers and the TOF enters and exits an ultra-high vacuum chamber via a
vacuum feed-through [93]. Inside the chamber (pressure ≈ 8 × 10−10 mbar), the
six-beam magneto-optical trap (MOT) produces a cold cesium atom cloud with a
1/e2 -diameter of 1.2 mm which is spatially overlapped with the nanofiber. Using
this setup, we are able to transfer the atoms directly from the MOT into the
trapping minima around the nanofiber just by appropriately changing the detuning
and the power of the MOT beams.
3.3.1
Loading of the trap
Optical dipole traps offer a nearly conservative potential which makes it necessary
to cool the atoms using dissipative light forces in order to transfer them into the
trapping sites. Alternatively, the trapping fields could be switched on abruptly in
order to capture atoms [76], however, this method never yields a higher atomic
density inside the dipole trap than in the MOT. In addition, it strongly depends on
the exact timing of both trapping lasers. We therefore employ a constantly present
trap, which is found to result in a much higher atom density than provided from
the MOT-cloud, thereby maximizing the number of trapped atoms.
Loading of our two-color trap is accomplished analogously to the procedure of
loading a focused beam dipole trap [95]: The red- and blue-detuned trapping light
fields are present in the nanofiber at all times. During the first 2 s the atoms are
captured and cooled in the MOT. In the following 80 ms the atoms are transferred
and cooled into the trapping minima along the nanofiber. For this purpose, the
power of the MOT cooling laser, repump laser and the magnetic field gradient are
3.3 Nanofiber trap
61
Figure 3.7: Timing sequence of the experiment. The colored curves show the
sequences of the powers and detunings of all laser used in our setup. The dashed
lines indicate the respective zero line. The magnetic field is ramped down after
loading the MOT.
ramped down to zero and the negative detuning of the cooling laser is increased to
about −80 MHz (see Fig. 3.7). We found that the high negative detuning of the
cooling beams is crucial for loading of our nanofiber trap. Because the cesium atoms
in the excited 6P3/2 state experience an inverted (repulsive) potential in the trap
(see Fig. 2.5), scattering of light from the MOT-beams leads to additional heating
thus to premature loss of atoms from the trap. Moreover, we find signatures of
polarization gradient cooling [86] in the MOT during the loading sequence, which
is found most efficient at high negative detunings and low intensities [96], resulting
in a temperature of about 1/6 of the Doppler temperature (see Sect. 3.4).
Loading efficiency
The optimal loading sequence has been found empirically and corresponds to exponentially decaying functions (∝ exp (−t/τ )) of all MOT parameters, with a common
time constant in the range of τ = 40 − 60 ms, yielding a maximal atom number
after about 100 ms of loading. For this purpose we monitored the loading procedure
by measuring the absorption of the probe beam for different decay time constants
τ . Here, the power of the transmitted probe light Pout ∝ Pin exp(−αNatoms ) decreases exponentially with the number of trapped atoms [15], thus the absorbance
(A ≡ − ln(Pout /Pin ) ∝ Natoms ) indicates the number of trapped atoms.
62
3,0
Absorbance (a.u)
2,5
2,0
1,5
1,0
0,5
0,0
1
10
100
1000
Loading duration (ms)
Figure 3.8: Loading curve of the nanofiber trap, measured from the absorption
of the probe beam as a function of loading duration. Each of the data points
have been taken for a loading time constant of τ = 50 ms and a different loading
duration (see text). The red solid line is a guide to the eye.
Figure 3.8 shows exemplarily the resulting loading curve for τ = 50 ms: After filling
the MOT for 2 s, the loading sequence has been applied with a fixed time constant
τ and the loading duration has been varied in order to find the optimal time to
interrupt this sequence. In the first 50 ms the absorbance and thus the number of
trapped atoms rises exponentially and reaches a maximum already after 80 ms. For
loading durations much longer than 100 ms the number of trapped atoms decreases
rapidly: The substantial change in the MOT-parameters for that long time leads to
an expansion of the MOT-cloud as well as to its acceleration under gravity. Hence,
loading from the residual cold-atom cloud is not efficient any more and does not
compensate the (light induced) losses of atoms.
In order to load as many atoms as possible, while keeping the temperature of the
trap very low, we thus apply the loading sequence for 80 ms. Note that by ramping
down the power of the repump laser the atoms get efficiently pumped into the lower
(F = 3) hyperfine ground state (> 99% after 100 ms), which decouples the atoms
from the cooling laser in order to reduce further heating due to scattering of light.
We further note that due to the small trapping volumes the loading is expected to
operate in the so-called collisional blockade regime resulting in an occupancy of at
most one atom per trapping site [97]. The resulting maximum average occupancy
of 0.5 in conjunction with the distance of about 500 nm between the standing
wave antinodes thus limits the maximum number of trapped atoms to 2000 per
millimeter.
3.3 Nanofiber trap
63
Figure 3.9: Saturation measurement. (blue circles) The absorbed power of the
resonant fiber-guided probe light has been measured for different values of the
input power. The asymptotic value of the absorbed power for strong saturation
indicates the number of trapped atoms. The red line is a theoretical fit, see text.
3.3.2
Atom number measurement
In order to determine more quantitatively the number of trapped atoms, we carry
out a saturation measurement: We tune the probe laser to the Stark-shifted resonance of the trapped atoms and measure the absorbed power as a function of the
incident power (see Fig. 3.9). At high saturation (s = I/Isat ≫ 1), the atomic
ensemble absorbs about PAbs ≈ 7.5 nW of probe light power. By comparing this
value with the power radiated by a single fully saturated cesium atom (cf. Sect. 2.1)
PCs =
Γ s
~ω0 ≈ 3.8 pW,
21+s
(3.4)
we infer that N = PAbs /PCs ≈ 2000 atoms are present in the fiber-based trap. The
solid red line in figure 3.9 is a theoretical fit based on a saturation model taking
into account the spatially varying intensity or power I(z) = P (z)/Aeff along the
atomic ensemble due to absorption
dP (z) =
n(z)σ0 /Aeff
P (z)dz,
1 + P (z)/Psat
(3.5)
where Psat = Isat Aeff is the saturation power, Aeff = I(r)/P the position dependent effective optical mode area of the evanescent field (see Sect. 1.3.2), σ0 the
atomic
absorption cross section on resonance, and n(z) the atomic line-density
R
( L n(z)dz = N ). The solution of this generalized Beer’s law, known as Wright
Omega function (or LambertW-function), can be easily evaluated numerically and
is used to fit the measured dependency of the absorbed power and the incident
power (i.e., PAbs (L, P0 ) vs. P0 ). The resulting saturation power Psat = 0.7(1) nW,
is consistent with the saturation intensity of the cesium D2-line at a radial distance
theo = 0.8 nW). For this theoretical value, we assumed orthogonal poof 230 nm (Psat
larization (σ-polarization), evenly populated MF states, and inhomogeneous broadening of 20 MHz of the atomic transition due to the ac-Stark shift (cf. Sect. 2.2).
64
3.3.3
Polarizations of the evanescent fields
The implementation of three dimensional trapping potentials around the nanofiber
requires an optimal control of the polarization at the waist of the TOF. Since the
TOF used in our setup is not intrinsically polarization maintaining, the state of
polarization of the dipole laser beams at the nanofiber waist needs to be carefully
aligned and monitored during the experiments. For this purpose, the angular distribution of the Rayleigh scattered light, which is emitted by the nanofiber, is observed
via two CCD cameras aligned perpendicular to the fiber axis. Small apertures of
the cameras (NA ≈ 1/140) ensure a high angular resolution (δα ≈ 5◦ ).
According to the radiation characteristics of dipole emitters, driven by linearly
polarized light I(ϕ) ∝ sin2 ϕ, with ϕ being the azimuthal angle with respect to the
direction of polarization, no light should be detected along the direction in which the
dipoles oscillate (ϕ = 0, π) (see Fig. 3.10 (a)). Since the HE11 mode carried by the
nanofiber exhibits spatially varying polarization (cf. Sect. 1.3.2), polarizers (PBS)
are placed in front of the cameras in order to separate the different polarization
components. By rotating the plane of polarization of the input beam a sinusoidal
modulation of the scattering intensities is expected, where the extinction ratio,
i.e., the ratio of minimum and maximum intensity (ER)= Imin /Imax , indicates the
degree of ellipticity of the polarization (see Fig. 3.10 (b)). Any deviation from
linear polarization of the fiber-guided light, e.g., induced by a birefringent TOF,
is indicated by a non-vanishing Imin , which can be compensated by using phase
retarders (e.g., Berek-compensators or a combination of λ/2 and λ/4-wave plates).
The line profiles of the imaged scattering along the nanofiber in Fig. 3.11 show
an almost uniform change of intensity with polarization angle ϕ. The variation of
the intensity along the nanofiber is caused by surface imperfections and density
fluctuations in the glass material. These randomly distributed scattering centers
inside the fiber lead to almost equal observed intensities of the z- and x-polarization
components. Without any separation of these components by using a polarizer, only
a small angular intensity variation is observable, presumably due to the azimuthal
dependence of the y-polarization component (see Fig. 3.10 (blue points)).
We note that the azimuthal phase dependency of the electric field component Ez
of the HE11 mode (see Sect. 1.3.2) gives rise to destructive interference of the zpolarized scattered light propagating along the y-direction (ϕ = 0, π), thus reducx
ing the minimal intensity of the x-polarization component Imin
to zero (for linearly
polarized fiber-guided light). The same argument holds for the electric field comz
ponent Ey , therefore also Imin
is expected to be zero for pure linear polarization.
The measured extinction ratios of the x- and z-polarization components reveal
an average residual ellipticity of 5% and 9% with and without pre-compensation,
respectively, which is compatible with the ellipticity of the beam at the output of
the TOF (2% and 5%) when accounting for the angular resolution of the cameras.
3.3 Nanofiber trap
65
(a)
cam 1
x
Dipole Emitter
y
z
cam 2
nanofiber
(b)
cam 1x
cam 2x
cam 1z
c1x+c1z
1.0
Intensity (a.u.)
0.8
0.6
0.4
0.2
0.0
0
90
180
Polarization angle
Figure 3.10: The angular dependency of Rayleigh scattering visualized as the
radiation pattern of a dipole emitter (a). Light which is polarized in the x-direction
is preferentially scattered in the perpendicular z- and y-directions. The nanofiber
provides field components in all three dimensions (see inset in (a)). The detected
intensities of the z- and x-polarization components (red and black line, respectively) of camera 1 and 2 (x-component, green line) for different polarization angles
ϕ are shown in plot (b), the solid lines are guides to the eye. The residual ellipticity of the light polarization is determined by the extinction ratio Imin /Imax .
Without polarizers being used the perceived intensity (blue dashed line) varies
only weakly with ϕ. The coordinate system is defined with respect to the polarization of incoming light. For this measurement laser light with a wavelength of
1064 nm has been used.
66
intensity (a.u)
20
15
10
5
0
0
2
4
6
8
position (mm)
10
12
Figure 3.11: (top) Imaged Rayleigh scattered light (λ = 1064 nm) emitted by
the nanofiber for parallel and orthogonal polarization (with respect to the image
plane) yielding minimum and maximum intensity respectively. The Plot below
shows the corresponding horizontal profiles for parallel (blue line) and orthogonal
(red line) polarization, respectively. The nanofiber waist ranges from 4 to 9 mm.
Similar results are obtained for simultaneous transmission of the red-detuned and
blue-detuned (1064 nm and 780 nm) lasers for the same configuration as used for
atom trapping. In spite of their relatively large powers and intensities, the birefringence of the nanofiber due to an optical Kerr effect or due to thermal influences is
negligible. Drifts of the ambient temperature, however, causes significant changes
of the polarizations. Nonetheless, the temperature stabilization in our laboratory
ensures a measured variation of the ellipticity of only a few percent over days.
3.3.4
Trap depth and -frequencies
For the trap configuration used throughout this work, the calculated trapping potential (see Fig. 2.3) with a trap depth of U0 = 0.4 mK yields a strongly confining
geometry in all three dimensions. This results in high oscillation frequencies of the
trapped atoms νr = 200 kHz, νz = 300 kHz, and νϕ = 140 kHz in the radial, axial,
and azimuthal direction, respectively. Here, the oscillation frequency νi of trapped
atoms is related to the curvature ∂i2 U (r0 ) of the potential (U (r) = U0 R(r)) at the
trap center r0 in the ith direction, which is equivalent to the ith spring constant of
a three dimensional harmonic oscillator, i.e.,
1
νi =
2π
q
U0 ∂i2 R(r0 )/m,
(3.6)
3.3 Nanofiber trap
67
1,0
Survival propabilty
0,8
0,6
0,4
0,2
0,0
0,01
0,1
Modulation frequency
1
(MHz)
Figure 3.12: Determination of the trapping frequencies. (squares) Measured
survival probability as a function of the excitation frequency ν, the black line is
a guide to the eye. The red curve results from a Monte-Carlo simulation of the
excitation process for our trap parameters.
where m is the atomic mass. Using the expression for the classical turning point
for atoms with a given energy E0 ,
p
xi = 2E0 /(2πνi )2 m,
(3.7)
we can estimate the localization of the atomic motion inside the trap.
Note that in the quantum mechanical picture, the localization of the atom is dictated by the uncertainty principle. For a harmonic oscillator the position uncertainty for minimal energy reads [5]
r
~
∆xi =
.
(3.8)
4πνi m
Hence, we expect a maximum confinement of atoms down to about ∆xr = 22 nm,
∆xz = 28 nm, and ∆xϕ = 33 nm along the axial, radial, and azimuthal direction,
respectively, in our trap.
Determination of the tapping frequencies
In order to verify the calculated trap geometry and to spot possible experimental
imperfections we performed a measurement of the trapping frequencies. Therefore,
we employed the method of resonant and parametric heat-out [6]. After loading
the atoms into the trap, the power of the red-detuned trapping laser is sinusoidally
68
modulated with a variable frequency using the AOM, thereby modulating the trap
depth and the position of the potential minimum [39]. A modulation depth of
10% of the laser power over t0 = 20 ms ensures a significant parametric (and
resonant) excitation of the atomic oscillation at the respective frequency. Note
that for a harmonic potential, resonant and parametric excitation would occur only
at frequencies νi and 2νi , respectively. This excitation heats up the atomic motion
and results in an increased loss rate of atoms.
In Fig. 3.12 we plotted the recorded survival probability, i.e., the relative atom
number N (ν, t0 )/N0 (t0 ) as a function of the excitation frequency ν, where N0 (t0 )
corresponds to the atom number measured at the time t0 without excitation. In
this plot, significant losses around ν = 180 ± 20 kHz and ν = 360 ± 50 kHz indicate
resonant and parametric excitation of the radial oscillation mode, respectively. A
small dip around ν = 520 ± 50 kHz exhibits weak parametric excitation of the
axial mode. The spectrum of parametric excitation of the azimuthal mode (around
280 kHz) is unresolvable because it coincides with the signatures of the radial
modes. Resonant excitation of the axial and azimuthal mode is negligible because
the power modulation does not change the position of the trapping minimum in
these directions. The measured trapping frequencies are in good agreement with
our calculations within the experimental uncertainties. Only deviations of about
10% to the low frequency side of the expected central frequencies and a slight
broadening are visible.
For a better understanding of the excitation and heating process in our trap, we
computed a theoretical excitation spectrum. The survival probabilities for different
modulation frequencies have been obtained from numerical Monte-Carlo simulations of the classical trajectories of thermal atoms in a time-varying potential of
the nanofiber trap (see Sect. 3.4). Therefore, the same parameters have been used
as for the measurement. Additionally, we assumed an initial temperature of 28 µK
of the atoms (see Sect. 3.4).
The resulting spectrum (see Fig. 3.12, red line) exhibits pronounced resonance
dips, similar to the experimental results, showing signatures of parametric and
resonant heating in our trap. The shift and broadening of these dips in both spectra
reflects the anharmonicity of our potential, which, however, is less pronounced than
observed in our experiments. We believe that the deviations from the theoretical
spectrum originate from small misalignments of the polarizations and power drifts
of the trapping lasers, which lower the trap depth and thus lead to reduced trapping
frequencies. Another explanation might be a small variation of the potentials along
the nanofiber. In this context, probing of the evanescent field along the nanofiber,
e.g., using a second nanofiber [41], could allow one to resolve the axial intensity
variation very precisely in this case.
3.3 Nanofiber trap
69
1.0
Rel. atom number (a.u.)
F=3
F=4
0.8
0.6
0.4
0.2
0.0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Time (s)
Figure 3.13: Evolution of the number of trapped atoms, pumped into the F = 3
and into the F = 4 state, as a function of time. The atom number decreases
exponentially ∝ exp(−t/τ ) with a state independent decay time (τ = 52 ± 4 ms),
revealing absence of cold collisions (see text).
3.3.5
Trap lifetime
The storage time of the atoms in the trap has been examined by measuring the
absorbance of fiber-guided probe light caused by the atoms for different delay times:
The change of the absorbance (for a detuning of ∆ = −20 MHz) of the probe light
was measured in discrete time steps for each individual experimental run. We find
that the absorbance, and thus the number of stored atoms, decreases exponentially
with a time constant of about 50 ms, smaller than what would be expected from
losses due to background-gas collisions for the measured pressure in our vacuum
chamber (≈ 4 s) [98] and from recoil heating due to light scattering (200 s).
The same short time constants have been found for a range of trap configurations,
i.e., varying the trap depth and position from 0.2 mK to 1 mK, and 200 nm to
350 nm distance from the surface, respectively. For much higher trap depths, the
cooling seems to be inefficient, which becomes apparent due to the low atom number
in the and the rapid atom losses.
At shorter distances from the fiber surface or lower potential depths, the confinement in the trapping sites is less robust against power fluctuation [39] which also
results in higher atom losses. Only lowering of the axial confinement, while keeping
reasonable trap depth by unbalancing our standing wave potential (Pred1 = 4.6
and Pred2 = 0.6 mW), leads to a small improvement of the lifetime up to 85 ms.
This result gives rise to the assumption that position fluctuations in axial direction, presumably due to phase fluctuations of the trapping fields, have a substantial
impact on the trap lifetime (see Sect. 3.5.4). For a further decrease of the axial
70
confinement, the lifetime reduces again to about 40 ms, because the atoms leave the
5 mm wide trapping region due to thermal motion along the nanofiber. We note
that in all relevant cases atom losses due to tunneling towards the fiber surface is
negligible because of the high repulsive potential barrier in our trap. Also surface
interactions are expected to be negligible at distances larger than 100 nm due to
the low conductivity of glass [105, 106].
We further note that by employing molasses cooling of the atoms during the storage
time, using the MOT-beams with a detuning of ∆ ≈ −80 MHz and relatively low
intensities (I/Isat ≈ 1), we are able to compensate heating and losses in our trap.
The maximal trap lifetime measured in fluorescence experiments (see Sect. 4.2) is
found to be τ = 3.5(2) s, which is expected to be limited only due to background-gas
collisions.
Cold collisions
In order to investigate the influence of hyperfine-changing collisions (HFC) on the
atom losses, and to estimate the number of atoms trapped in each potential well,
we performed the lifetime measurements for atoms prepared in the F = 4 and the
F = 3 hyperfine ground state, respectively. Since atom losses due to HFC occur if at
least one of the colliding cold atoms is in the energetic higher hyperfine ground state
(i.e., F = 4) [99]; the energy released from the collision-induced F = 4 → F = 3
transition suffices to expel the atoms from the trap. For the low volume of each
trapping sites and high oscillation frequencies these collision between cesium atoms
would lead to rapid atom losses (< 100 µs) [99]. Collisions within a pure F = 3
population, however, should not show any significant influence on the trap lifetime
[6]. We therefore optically pumped the trapped atoms, while loading them into the
nanofiber trap, into one of the desired states with high purity (& 99%), and checked
that the population stayed unchanged until each measurement was performed. For
probing of the “dark” F = 3 population the atoms were repumped to the “bright”
F = 4 state after the respective delay time using the repump beams of our MOT.
The evolution of the atom number of the trapped ensemble, pumped into the F = 3
and into the F = 4 state, are shown in Fig. 3.13. We find that the lifetime of the
trapped atoms is independent of the hyperfine ground state population. Thus, we
can conclude that no such HFC collisions between the atoms occur after loading
the trap. Furthermore, this result confirms that an occupancy of at most one atom
per trapping site is already reached due to light-induced collisions caused by the
MOT-beams during the loading procedure.
With regard to the negligible cold- or background gas collisions, heating is expected
to be the dominant loss mechanism in our trap. This will be the subject of the next
sections.
3.4 Temperature measurement
3.4
71
Temperature measurement
Since the temperature is determined by the balance of heating and cooling processes,
the detailed study of the temperature of the atoms trapped in the evanescent field of
a nanofiber might reveal possible technical or physical origins of heating and limits
of cooling mechanisms. Moreover, the measurement of the temperature provides
information about the confinement of atoms in our trap, which is of importance in
regard to the interaction of the atoms and light. For example, the thermal motion
of the atoms in a dipole trap leads to inhomogeneous broadening of their transition
frequencies due to spatial dependent light shifts. In addition, the delocalization with
respect to the propagation direction of the fiber-guided light leads to dephasing of
the atoms with respect to the probe field, and thus limits the coherence time of
light-atom interactions.
Method
The temperature of a small sample of cold atoms is not directly accessible, e.g.,
by contact measurement, but can be deduced from the position, momentum or
energy distribution. A time-of-flight measurement as it is used for the temperature
measurement in, e.g., focused beam traps [6], cannot be applied because of the
atoms’ close proximity to the fiber surface. Thus, we will follow the approach of
adiabatic lowering of the trap reviewed by W. Alt [100]. This method relies on a
slow lowering of the trap depth from initial value U0 to Ulow such that only the
atoms with energies smaller than a certain threshold E will remain trapped. For
such an adiabatic lowering, the threshold energy E is only dependent on the final
trap depth Ulow . By an appropriate change of Ulow the energy threshold E can
be varied in order to find the cumulative energy distribution, out of which the
temperature can be estimated.
The adiabadicity criterion is to change the dipole potential on a time scale longer
than the largest oscillation period. An optimized time sequence of U (t) for an
anharmonic potential is found by [100]


for t = 0
U0






 √
t2
U
1
−
for 0 < t < Tc 2
U (t) =
(3.9)
0
2
4T
c






√

U Tc2
for t > Tc 2
0 t2
with the critical time Tc ≈ 10 × 2π/Ωϕ = 0.1 ms. In view of the high anharmonicity
of our trapping potential we further reduced the lowering time by a factor of 10
(see Fig. 3.14).
72
(a)
1,0
U(t) / U0
0,8
0,6
0,4
0,2
Ulow
0,0
0
1
2
3
4
5
time (ms)
(b)
0.0
E=U
potential (mK)
esc
0.1
0.2
U
0
0.3
E
0.4
100
200
0
300
400
500
600
700
radial distance (nm)
Figure 3.14: (a) Time sequence U (t) optimized for adiabatically lowering of
the trap depth. (b) By lowering the potential depth from U0 to Ulow the atomic
energy E decreases from the initial value E0 due to adiabatic cooling. Since E
decreases slower than U the atom escapes from the trap if the energy reaches the
potential depth Uesc .
73
The adiabatic reduction of the depth of a “conservative” potential does not conserve
energy, but it is rather the action integral over one oscillation period,
I p
I
√
E − U (x) dx,
(3.10)
S = p dx = 2m
which is conserved [103]. In this case, it can be shown that the energy E of the
atom is reduced due to “adiabatic cooling”, but this is less than proportionally
with respect to U . Hence, at a certain depth Uesc , when the energy E reaches the
threshold Uesc , the atom escapes from the trap. The threshold energy Eesc = Uesc ,
in this case, is lower than the initial energy (Eesc < E0 ) (see Fig. 3.14 (a)). As a
consequence, the measurement of the energy distribution in this way is inherently
affected by adiabatic cooling, which has to be considered in our experimental results.
Experimental realization
The experimental procedure of the measurement of the temperature was performed
directly after loading about 2000 atoms into the trap with an initial trap depth
of U0 = 0.4 mK. The potential was lowered afterwards according to Eq. (3.9) by
changing the power of the red-detuned laser using the AOM. In order to detect the
fraction of “surviving” atoms we measured the absorption of a detuned light pulse of
0.5 ms length before lowering the trap and after restoring the original potential with
the same radial distance to the fiber. The restoring ramp does not have to follow the
adiabatic form, since no atoms are lost upon increasing the trap depth. A low power
of the probe light ensures that the scattering rate of the atoms (30 kHz) is small
compared to their oscillation frequency in the trap (& 140 kHz), thereby reducing
the effects of recoil heating [127]. By this means, we find a loss rate of less than 10%
after 5 ms without any lowering of the potential. The measurements for differently
lowered potential depths (Ulow ) result in an altered energy distribution, which needs
to be corrected by transforming Ulow into the corresponding initial energy E0 . Since
no analytic expression is known accounting for adiabatic cooling in the anisotropic
potential of our trap, we performed a numerical Monte-Carlo simulation in order
to find the adequate relation between Uesc and E0 (see Fig. 3.15).
Numerical simulation
For estimation of the effects of adiabatic lowering of our trap (and to estimate the
effects of parametric and resonant heating (see Sect. 3.3.4)), we have carried out
a numerical simulation of the motion of atoms inside the three dimensional timevarying potential of our nanofiber trap V (r, ϕ, z, t) = U (t)R(r) cos(2ϕ) cos 2 (βz) +
(v.d.W+gravity) (cf. Sect. 2.3). For that purpose, the classical equations of motion
have been solved numerically using the Kutta-Merson algorithm [104], implemented
into a C++ programm.
74
1.0
0.8
E0/U0
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Uesc/U0
Figure 3.15: Initial energy E0 as a function of the escape depth Uesc . The data
points are obtained from a Monte-Carlo simulation of 10, 000 trajectories in the
3D-potential well of our trap. The error bars indicate the escape range. The red
line represent a fit using an analytic function (y = axb + cxd ).
This simulation of the atomic trajectories has been performed under the same
conditions (i.e., trap depth and lowering function U (t)) as the measurement. In
order to enhance the calculational speed the radial potential function R(r) was
approximated as a sum of two exponential functions fitted to the exact solutions
of the red- and blue-detuned fields. In this program, the chosen accuracy (10−3 )
is compatible to a step time of 3 ns. The atomic trajectories for different initial
energies E0 are calculated as U (t) is varied according Eq. (3.9). The final depth
Ulow is held for 1 ms such that all unbound atoms have enough time to leave the
trap. If the atoms displace reasonably far from the trap center, i.e., more than 2 µm,
π/2 and 250 nm in radial, azimuthal and axial direction from the trap minimum,
respectively, they are considered as lost. This procedure is repeated many times for
random starting positions (with respect to E0 ) and phases of the atomic oscillation
in order to find the probability p(E0 , Ulow ) of an atom remaining trapped for a
certain E0 and Ulow . The obtained probability distribution p(E0 , Ulow ) runs from
unity to zero for an decreasing Ulow . Here, the escape depth Uesc is defined as
the lowered depth (Ulow ) for which p(E0 , Ulow ) = 0.5, the according escape range
∆Uesc defines the energy resolution. Both quantities are obtained by fitting an
error-function erf{(Ulow − Uesc )/∆Uesc } on p(E0 , Ulow ).
The resulting relation between Uesc and E0 , evaluated from eleven sets of simulations with different E0 , is plotted in Fig. 3.15. The non-linear dependency of the
escape depth on the initial energy, where E0 is larger than Uesc , suggests adiabatic
cooling for atoms with energies E0 < 0.7U0 . For atomic energies approaching the
trap depth U0 lowering of the potential according Eq. 3.9 in our simulations results
in raising of the escape depth (i.e, Uesc > E0 ), probably because lowering of the
trap in this case is not adiabatic anymore.
75
1.0
1.0
0.8
0.8
= 0.91
P1
kT /U0 = 0.07
0.6
0.02
0.002
0.6
0.4
0.4
0.2
0.2
0.0
0.0
0.2
0.4
0.6
0.8
probability density (a.u)
fraction of atoms with
energy below E0
0.0
1.0
E0/U0
Figure 3.16: Cumulative energy distribution of the trapped atoms. The data
points are obtained from the measured survival probability as a function the lowered potential Ulow . The former dependency on Ulow is transformed into E0 dependency using the results from the Monte-Carlo simulation in Fig. 3.15. The
fit of the cumulative Maxwell-Boltzmann distribution (red line) in our trap yields
a temperature of 28 µK (see text). The dotted line indicates the corresponding
energy distribution.
For further utilization of the results obtained from the simulations, an analytic
function was fitted to the data points. This function was applied to the experimental data in order to extract the true (cumulative) energy distribution from our
measurement (see Fig. 3.16).
Results
The corrected cumulative energy distribution (see Fig. 3.16), i.e., the fraction of
atoms with an energy below E0 , is well described by the integrated Boltzmanndistribution
Z E0
g(E) −E/kB T
P (E0 , T ) = Pmax
e
dE,
(3.11)
Z(T )
0
where
g(E) =
2π(2M )3/2
h3
Z
V (E)
p
E − U (r)d3 r
(3.12)
denotes the density of states [107], and Z(T ) is an appropriately chosen normalization function, where kB is the Boltzmann constant and T is the temperature. For
E ≪ U0 the latter integral yields approximately
g(E/U0 ) ∝
E2
E4
E6
+
+
O(
).
2U02 4U04
U06
(3.13)
76
Using this approximation, the resulting fit of P (E0 , T ) (Eq. (3.11)) on our data
yields a temperature of kB T = 0.070(2) U0 and a maximum survival probability of
Pmax = 91%. For the calculated trap depth of U0 = 0.4 mK we obtain a temperature
of T = 28 ± 1 µK. This value is lower than the Doppler temperature (125 µK), and
even lower than what would be expected when taking evaporative cooling solely
into account (T ≈ 1/9 U0 ) [87, 108] (see Sect. 3.1). Hence, we can conclude that
sub-Doppler cooling mechanisms [5, 86] may operate even in the presence of the
nanofiber and the trapping fields, which might indicate a viable way of further
cooling the atoms.
By knowing the temperature, we can estimate the occupation of the vibrational
states and the localization of the atoms inside the trap. The mean occupation
number is defined by [110]
1
hni i = ~Ω /k T
.
(3.14)
i
B
e
−1
With the measured trapping frequencies νi (= Ωi /2π) from Sect. 3.3.4, we obtain
hnz i = 1.4, hnr i = 2.5, and hnϕ i = 3.8. These low mean occupation numbers
indicate that the atoms can be found very close to the vibrational ground state of
the respective oscillatory mode. The corresponding extension of the atomic motion
√
in each spatial direction, estimated from the 1/ e-widths of the harmonic-oscillator
wave function modulus squared,
2
X p
2
|Ψ(xi )|2 ∝
(3.15)
Hn ( mΩi /~xi )e−mΩi xi /2~ e−n~Ωi /kB T ,
n
where Hn are the Hermite polynomials, yields only about twice the spatial extension
of the oscillatory ground state wave function in each direction, i.e., ∆z ≈ 42 nm,
∆r ≈ 67 nm, and r∆ϕ ≈ 92 nm for the axial, radial and azimuthal direction,
respectively.
3.5
Heating-rates
The method of temperature measurement described in the previous section is used
now in order to estimate the total heating rate of the atoms during the storage
in our trap. We measured the temperature for different delay times after loading
the trap and switching off the MOT-beams. Figure 3.17 shows the progression
of the temperature in the first 100 ms, starting at T ≈ 0.07U0 = 28 µK and
converging towards a maximum value. During this time the number of atoms
reduces exponentially to 13.5% according to N = N0 exp (−t/50 ms). The increase
of temperature in the first 100 ms resembles a similar time dependency; the fit
of a simple exponential model T (t) = Tmax − T0 exp (−t/τ ) on the measured data
points indicates that both follow the same e-folding time of τ ≈ 50 ms. The
correlation of the time evolution of the temperature and the time evolution of the
3.5 Heating-rates
77
0,13
T(t) = T0*exp(-t/ ) + Tmax
norm. temperature kBT/U0
0,12
0,11
Tmax
0.12
T0
-0.05
59
0,10
0,09
0,08
0,07
0,06
0
20
40
60
80
100
120
time (ms)
Figure 3.17: (black points) Measured temperature as a function of storage time.
The red line is a fit (see text). Note that the increase of the temperature is accompanied with an exponential decrease of the atom number N = N0 exp (−t/50 ms).
atom number may suggest that atom losses arise due to heating inside the trap,
which is found to be more pronounced than what one would expect from simply
considering scattering of photons from the trapping fields. In the following section
several sources of heating are discussed and their contributions are evaluated. The
analysis of the fundamental and technical heating effects in our trap is intended to
quantify heating- and loss-mechanisms arising for the storage of the atoms.
3.5.1
Recoil heating
One fundamental heating mechanism in an optical dipole trap arises from spontaneous scattering of photons of the dipole fields. The absorption and emission of a
photon transfers each on average the recoil energy
Er =
(~k)2
2m
(3.16)
to the atom [6], where k denotes the wave number of the photons. For a random
momentum transfer at the rate Rs the average atomic energy increases by [6]
hĖi = 2Rs Er .
(3.17)
For a potential depth U and detuning of the dipole laser frequency (∆ ≫ Γ) the
Γ
scattering rate is given by Rs = U ~∆
(cf. Sect. 2.1). Due to the large detunings
of both of our dipole trap lasers and a trap depth of U/kB = 0.4 mK, only a low
scattering rate of Rs = 10 s−1 is expected, producing a negligible heating rate of
4 µK/s.
78
E
F2
|2i
F1
|1i
x
Figure 3.18: Illustration of heating induced by dipole force fluctuation. The
dipole force changes from F1 to F2 upon excitation from state |1i to |2i
3.5.2
Heating due to dipole-force fluctuations
Another form of heating arises if an atom undergoes a transition from the ground
state |1i to the excited state |2i. In this case, the atom experiences a repulsive potential during its lifetime in state |2i (see Sect. 2.2). Several such random excitations
result in fluctuating dipole forces which lead to heating of the atomic motion. For
oppositely curved potentials the dipole force is changed by ∆F = ∇U1 (r) − ∇U2 (r)
for the lifetime τ = 1/Γ of the excited state. This force transfers momentum of
∆p = ∆F τ in the stationary case, i.e., if the oscillation period Tosc = 2π/Ω is
large compared to τ (30 ns), as in our case. In analogy to recoil heating, these
momentum kicks drive random walks in momentum space. The energies of these
kicks averages to
(∆p)2
hEkick i =
,
(3.18)
2m
and result in the heating rate
hĖi = Rexc hEkick i .
(3.19)
For a real atom this effect is, however, state selective in the hyperfine- and Zeeman
sub-levels (cf. Fig. 2.5 in Sect. 2.3). In our case, the maximum energy of hEkick i =
3 µK is obtained for the excitation to the |6P3/2 , F = 5, M = 0i state, assuming
an thermal energy of E = 0.5 U0 of the atom. Note that for atoms excited to the
6P1/2 states hEkick i is negligible small (see Sect. 2.2).
In the case of off-resonant scattering of the trap light the excitation rate is given
by [63]
R2
Rexc = s .
(3.20)
Γ
The scattering rate in our trap yields Rexc ≈ 10−6 s−1 , which results in a completely
negligible heating rate below 10−9 mK/s.
3.5 Heating-rates
79
If, however, near-resonant light interacts with the trapped atoms, e.g., for optical
pumping or for probing of the atoms, then the excitation rate becomes equal to
the scattering rate, which can be significant. As an example, the typical power
of our resonant probe beam of 1 pW yields an excitation rate of Rexc = 30 kHz
to the 6P3/2 state. For an exposure time of 0.5 ms, as used for the temperature
measurement, the temperature increases by about 6 µK. Here, the energy dependent
heating rate changes the temperature by 2 µK, 3 µK, 1 µK in the radial, axial and
azimuthal direction, respectively. Thus, in order to avoid the excitation of higher
vibrational modes in the trap, the pulse length ∆t or the number of scattered
photons (Nphot = ∆tRsc ) should be kept small.
In general, resonant excitation can be fully suppressed during the storage time
by careful shielding of the experimental setup against resonant scattered light.
However, for our case, the blue-detuned laser light propagating inside the nanofiber
induces fluorescence of the glass material, producing about 10−4 pW of (near)
resonant light in a frequency band of ∆ω = 2Γ. The corresponding scattering rate
is estimated to be at a level of Rexc ≈ 1 s−1 , which gives a lower limit of our heating
rate of about 3 µK/s.
3.5.3
Potential fluctuations
Technical noise induced by the lasers themselves or the control units of the AOMs
can have a strong impact on the intensity or phase stability of the trapping light
fields. Any variations of these parameters are changing the geometry of trapping
potentials. Fluctuations in the potential depth, in the Fourier domain at twice the
trapping frequency, affect the amplitude of the oscillatory motion, an effect that is
referred as parametric heating. Position fluctuation of the potential minimum in
resonance with the trapping frequencies, strongly influences the atomic oscillation as
well. Single fluctuations of both kinds can either increase or decrease the oscillatory
energy of a given atom. However, the stochastic nature of these fluctuations always
leads to heating.
Parametric heating
Because of the power dependency of the trap depth U0 ∝ P , fluctuations of the laser
power result in fluctuations of the trap depth and therefore cause fluctuations in the
spring constant of the trap. This parametric heating effect is thus determined by
the amplitude and the spectral distribution of the laser power fluctuations ∆P =
P (t) − hP i around hP i. The only relevant contribution of these fluctuations to
heating is found in the spectral region close to twice the trapping frequency (2νi ).
Any excitations at other frequencies will dephase with respect to the oscillatory
motion of the atom and will thus on average not lead to heating.
80
An important quantity concerning these processes is thus the spectral density of
the relative power noise SP (ν), which is related to the normalized variance of the
power or the potential in a spectral band dν around the frequency ν by
∆P 2 dν
∆U 2 dν
=
(3.21)
SP (ν) dν =
hP i2
hU i2
and thus
Z
0
∞
SP (ν) dν =
∆P 2
hP i2
=
∆U 2
hU i2
.
(3.22)
Here and in the following, the notation ν(= ω/2π) for the frequencies in units of
Hertz is used.
For a harmonically approximated potential U (xi ) ∝ x2i and small variations of the
power (∆P ≪ hP i) at the frequency 2νi , the classical as well as the quantum
mechanical calculation of the equation of motion yield an exponential growth of
the mean energy in time [108, 109]
hĖi = π 2 νi2 SP (2νi ) hEi .
(3.23)
Note that the quadratic term νi2 makes parametric heating most efficient at high
oscillation frequencies, i.e., strongly confining trap geometries are very sensitive to
heating due to power fluctuation.
For our two-color dipole trap the sum of the two individual potentials results in
a potential well with a depth and radial position that strongly depends on the
relative values of Pred and Pblue . Small variations of these powers have a stronger
effect on the total potential than on each individual. More quantitatively, the power
fluctuations of the respective lasers results in a factor cred ≈ 3 and cblue ≈ 2 times
higher potential variation as compared to the respective individual potentials.
We recorded the intensity noise spectra or the spectral variance of both lasers
at the output of the TOF using a fast photodiode and an electronic spectrum
analyzer (Agilent, E4407) (see Fig. 3.19). The relative intensity noise is obtained
by dividing the spectrum by the DC output voltage. At low frequencies the 1/f noise is apparently as well as several peaks from the switching power supplies of the
lasers or relaxation oscillations. In the relevant region from 100 kHz to 1 MHz the
spectrum is essentially flat and reaches the shot noise limit of the DC photo current.
The obtained noise level is thus only an upper limit. Integration of the power
spectrum Sred (ν) of the red-detuned laser yields a slightly higher rms noise level of
0.3% than specified by the manufacturer (< 0.2%). For the blue-detuned laser the
measured rms noise level yields 1%. Using Eq. (3.23), we estimate the maximum
heating rate for the measured noise of the individual laser and the combined total
value (hĖtot i = cred hĖred i + cblue hĖblue i), respectively (see Fig. 3.19(b)), assuming
an energy of E = U0 /2. The maximum heating rate is expected for the axial mode
(2νz = 600 kHz), yielding only a negligible value of hĖi = 10 µK/s.
3.5 Heating-rates
81
(a)
rel. intensity noise
(spectral variace) (1/Hz)
1E-11
1E-12
1E-13
1E-14
1E-15
1E-16
1000
10000
100000
1000000
1E7
frequency (Hz)
(b)
100
parametric heating rate (U0/s)
10
1
0,1
0,01
1E-3
1E-4
1E-5
1E-6
1E-7
1E-8
1E-9
10.000
100.000
oscillation frequency 2
1.000.000
1E7
(Hz)
0
Figure 3.19: (a) Spectral density of the relative intensity noise of our dipole
lasers transmitted through the TOF, the spectra for the blue-detuned and reddetuned lasers are labeled by the color of the line. The green line indicates the
noise of the amplified photodiode (lasers off). (b) Resulting parametric heating
rate as a function of the double oscillation frequency of the individual lasers (red,
and blue line) according to Eq. (3.23), and the total heating rate (black line). The
grey dashed lines mark twice the radial and axial trapping frequencies.
82
Resonant heating
The contribution of resonant heating due to position fluctuation of the trap center
is obtained analogously to Eq. (3.23) by [109]
hĖi = 4π 4 mνi4 S∆r (νi ) ,
(3.24)
where S∆r is the power spectrum of the position fluctuation, i.e., integration of S∆r
over the frequency space yields the variance of the trap center position
Z ∞
S∆r (ν) dν = ∆r 2 .
(3.25)
0
Here, a shaking of the trap heats up the atoms independently of their initial energy,
but with a strong dependency on the trap frequency due to the factor νi4 .
Since variations of the relative powers Pred and Pblue result in radial displacement
of the trap center, fluctuations in the power of each dipole laser give rise to trap
position fluctuations in the radial direction. For small power fluctuations ∆P , the
variance of the trap position can be approximated
∂r 2
h∆P 2 i ,
h∆r i ≈ ∂P 2
such that
Z
0
2
∂r
2
∂P SP (ν)dν = h∆r i .
∞
(3.26)
(3.27)
For our trap parameters, the calculated power dependency of the trap position
yields for both lasers ∂r/∂(P/P0 ) ≈ 2 × 10−7 m/(P/P0 ), where P0 is the power of
the respective laser beam. Hence, the measured power noise spectrum SP (ν) can
be used in order to estimate both the parametric and the resonant heating rate in
our two-color dipole trap.
For the measured spectral power densities SP (νr ) at the radial trap frequency νr =
200 kHz (see Fig. 3.19), Eq. (3.24) predicts a total resonant heating rate of hĖi =
0.2 mK/s in our trap. In this analysis, we find that the (radial) resonant heating
rate is expected to be significantly larger than the (radial) parametric heating rate
for the initial temperature of 28 µK. This is also reflected in the excitation spectrum
obtained from measurement of the trapping frequencies (see Fig. 3.12). However,
since the power fluctuations of our dipole lasers are small at the relevant frequencies
(see Fig. 3.19), atom losses due to this kind of resonant and parametric heating can
be neglected.
Note that vibrations of the fiber itself might also provide resonant heating, since the
frequencies of normal modes of vibrations of the nanofiber waist (νn ≈ n · 130 kHz)
3.5 Heating-rates
83
/4
/4
PBS
PBS
Figure 3.20: For the measurement of the phase noise we used the Sagnac interferometer structure of our setup. The interference of the two beams is detected
using a fast photodiode (PD). The resulting beat signal is recorded by an electronic spectrum analyzer. Two AOMs with mutual detunings are used for the
heterodyne measurement.
are close to the trapping frequencies. Preliminary measurements of the phase fluctuations on several other TOFs indeed indicate resonances of mechanical oscillations.
However, it is unclear whether excitations of these high frequency oscillations do
occur in our setup. Currently, the measured phase noise spectra in our setup are
dominated by a broad noise floor (above shot noise limit) which does not permit
to resolve resonance features of any mechanical vibrations.
3.5.4
Influence of phase noise
The periodic lattice potential of the nanofiber trap relies on the interference of two
counter-propagating red-detuned laser beams. This interference forms a standing
wave intensity pattern which is strongly dependent on the coherence properties of
the laser light as well as on the phase stability of each of the two counter-propagating
beams.
Typical coherence lengths provided by single-frequency solid state lasers are in the
order of many kilometers. In free space, it is only determined by the laser line
width γ as Lcoh ≃ c/γ. In general, the coherence length is limited by phase noise,
e.g., resulting from spontaneous emission in the gain medium, or noise induced by
electronics when using modulators, e.g., AOMs. If, however, intense laser light is
carried by optical fibers, non-linear effects give rise to phase fluctuations induced by
stimulated Brillouin scattering (SBS). This effect reduces the coherence length and
results in intensity and phase noise due to stochastic fluctuations of the spontaneous
scattering processes that initiate SBS [111].
84
1E-10
2
phase noise (rad /Hz)
total phase noise (2 AOMs)
optical phase noise
1E-11
1E-12
1E-13
0,0
200,0k
400,0k
600,0k
800,0k
1,0M
frequency (Hz)
Figure 3.21: Spectral density of the optical phase noise measured with the
setup shown in Fig. 3.20 (red line, no AOM), (black line, 2 AOMs on).
Brillouin scattering
SBS can be substantial in long haul single mode fibers resulting from the interaction
of the transmitted light and thermally generated acoustic waves in the fiber. Above
a certain threshold power, which can be as low as a few milliwatts for narrow
linewidth lasers, the scattered light interferes with the incident light and amplifies
the acoustic waves via electrostriction [112]. This further enforces the scattering
process and results in a large back reflection with a Stokes-shifted frequency (about
eff
20 GHz). However, the necessary threshold power for SBS is Pthr ≈ 21 AgL
[113],
−11
m/W is the gain coefficient for silica, Aeff is the effective fiber
where g = 2 × 10
cross section, and L the length of non-linear interaction. For the nanofiber waist the
SBS threshold is thus expected to be about Pthr ≈ 10 W, hence only spontaneous
Brillouin scattering should occur in this case. Experimentally, we have not observed
any non-linear dependency of the backscattered power due to SBS up to a level of
250 mW of transmitted light through the TOF in vacuum. Only the analysis of
phase fluctuations in our optical setup (see Fig. 3.21) may give rise to spontaneous
Brillouin scattering and to heating due to axial position fluctuation of the trap.
Note that the optical path lengths of the two counter-propagating laser beams are
almost equal in our setup. Thus, common phase noise of the standing wave beams
at the nanofiber is expected to be strongly suppressed. For unequal path lengths,
the intrinsical phase noise of the laser itself should have an effect on the heating
rate and thus on the trap lifetime. However, even for a difference of ∆L = 2 m
in the optical pathways we do not observe any change neither of the phase noise
nor on the trap lifetime. Hence, we can conclude that the contribution of phase
fluctuations produced in the laser cavity and in the optical pathways to heating is
negligible.
3.5 Heating-rates
85
Phase noise measurement
For the measurement of the phase noise of the standing wave we employed the
Sagnac interferometer structure of our fiber trap setup (see Fig. 3.20). The imperfections of the polarizing beam splitter (PBS) are used to create interference of the
two counter propagating laser beams at the output port of the PBS (see Fig. 3.20).
In a homodyne detection of the interference (or via heterodyning using two AOMs),
using a fast photodiode (PD), we measure a phase dependent intensity signal
I(φ + ∆φ) ∝ I0 (1 − cos(φ + ∆φ)).
(3.28)
Any deviation of the relative phase φ of the counter propagating beams by ∆φ
results in an intensity change at the PD. The resulting PD signal is recorded and
processed using an electronic spectrum analyzer (ESA). The highest sensitivity for
phase fluctuations in this setup is obtained at the side of the fringe (φ = π/2),
provided that slow phase drifts are negligible during the measurement. From the
determined intensity noise we estimate the phase noise via the relation
∆φ = |∂φ I|∆I.
(3.29)
When using two individual AOMs for each of the dipole beams in the setup, we
measure a phase noise density of S(ν)∆φ ≈ 1 × 10−11 rad2 /Hz with a mainly flat
spectrum from 400 kHz to 1 MHz, at lower frequencies the 1/f -noise is governed
by the electronic noise of the AOM-drivers (see Fig. 3.21). Without any AOMs
the phase noise level is reduced to an almost constant level of S(ν)∆φ ≈ 2.2 ×
10−12 rad2 /Hz in the region from 200 kHz to 1 MHz, the corresponding phase
variation ∆φrms decreases from 23 mrad to 1.5 mrad (DC to 1 MHz).
These phase fluctuations of are related to axial position fluctuations of the trap
center via
∆zrms = ∆φrms /β,
(3.30)
where β is the propagation constant of the red-detuned dipole laser. Hence, phase
fluctuations of the standing wave lead mainly to resonant heating in axial direction.
For an axial oscillation frequency of νz = 300 kHz, Eq. (3.24) predicts a heating
rate of hĖi ≈ 68 U0 /s = 27 mK/s (or 5 mK/s without any AOMs). This value is
compatible with the reduced lifetime of τ ≈ 12 ms resulting from the additional
electronic noise induced by the AOM controllers. Without any AOMs being used,
the residual phase noise would yield a heating rate of 5(1) mK/s, which is in a good
agreement with the measured loss rate of atoms (U0 /τ ≈ 8(1) mK/s) for τ ≈ 50 ms.
Note that we are able to slightly reduce the atomic losses by decreasing the axial
trapping frequency, i.e., by lowering of the axial confinement only (cf. Sect. 3.3.5).
This behavior would suggest that position fluctuations of the standing wave beams
contribute significantly to heating in our trap. The origin of the position fluctuations is assumed to be phase noise due to spontaneous Brillouin scattering inside the
86
TOF. However, further investigations of this effect need to be carried out in order
to clarify the underlying mechanisms and eventually suppress the arising heating.
Results and discussion
The survey of several sources of heating reveals that most of the heating mechanism
are negligibly small (see Table 3.2). The only significant contribution, which also
seems to be the origin of the reduced trap lifetime, is indicated by phase fluctuations. The underlying mechanism is not yet fully understood and requires further
investigations. Nonetheless, we hope that the phase of the trapping fields can be
actively stabilized by using an electro-optical phase modulator and employing a fast
servo loop. This would reduce heating and increase the storage time of the atoms,
and would even allow us to efficiently convey the atoms along the fiber.
We further note that any kind of heating reduces the coherence time of a quantum
system, either due to dephasing by thermal motion or, e.g., due to loss of atoms,
etc.. However, the timescale on which inevitable photon-scattering of the dipole
laser fields leads to decoherence, in our case about 50 ms, is comparable to the trap
−1
−1 −1
+ τloss
) ,
lifetime. Thus, the coherence time of the atomic ensemble τcoh ≈ (τscatt
is expected to be reduced to only half of the ultimate value.
Although, heating limits the lifetime of our trap, the effect on the coherence time of
the trapped ensemble is assumed to be only marginal (compare with [29]). Hence,
our optical interface with trapped atoms holds great potential for coherent manipulation of atomic states with light. It should thus be possible to, e.g., realize
electro-magnetically induced transparency (EIT) and slow-light in our system, a
first important step towards fiber coupled quantum memories for quantum communication.
Heating effect
Recoil heating
Dipole force fluctuation
(due to fluorescence)
(due to probe light)
Laser intensity fluctuation (total)
Laser intensity fluctuation (radial)
Optical phase noise (axial)
Phase noise (AOM)
temperature change
estimated from atom losses
Heating rate
10 µK/s
< 10−6 µK/s
3 µK/s
(12 µK/ms)
10 µK/s
0.2 mK/s
5 mK/s
(27 mK/s)
Comment
fundamental
state dependent
and energy dependent
parametric, energy dependent
resonant
resonant
resonant, avoidable
≈ 8 mK/s,
(≈ 33 mK/s)
observed
(two AOMs)
Table 3.2: List of heating effects and the corresponding heating rates.
3.6 Conveying atoms
87
Figure 3.22: The mutual detuning ∆ν of the counter-propagating beams results in a standing wave pattern which moves at a velocity of v = λ/2 · ∆ν. In
a co-moving frame this detuning is compensated by the Doppler shift and the
interference forms a standing wave at rest.
3.6
Conveying atoms
As a first straightforward application of our nanofiber-based trap, we considered
the realization of an optical conveyer belt for atoms. In this section I will report on
the first attempts to load and to transport the trapped atoms along the nanofiber.
3.6.1
Moving standing wave
If the axial confinement in an optical dipole trap is provided by a standing wave,
the stationary optical lattice can be translated by varying the relative phase of
the counter-propagating beams. A constant motion of the lattice can be realized
through a mutual detuning of the frequency of the beams such that the interference
pattern propagates with a velocity of
v=
λ∆ν
,
2
(3.31)
where ∆ν is the frequency difference of the counter-propagating beams (see Fig. 3.22).
In a co-moving frame this detuning is canceled by the Doppler shift thereby resulting in a standing wave pattern. For small accelerations the trapped atoms will
remain trapped, allowing the atoms to be conveyed along the fiber [114, 115].
The detuning of both lasers can be realized by using acousto-optical modulators
(AOMs). However, as discussed in the previous section, the lifetime of the trap
critically depends on the relative phase stability of the counter-propagating beams.
By employing further active optical elements, such as AOMs, additional phase noise
might be introduced, which results in higher heating rates and leads to a rapid loss
of atoms. Due to the sensitivity to phase noise, efficient transportation requires an
active stabilization of the phase, which at the same time has to allow for tuning of
the laser frequencies in order to accelerate the atoms.
88
Figure 3.23: Conveying of atoms. The images a)−e) have all been recorded
after loading of traveling potentials moving with different velocities v = λ/2 · ∆ν.
The respective detunings ∆ν of the counter propagating beams are denoted on
the right.
3.7 Imaging of the trapped atoms
3.6.2
89
Transport of atoms
For a simple proof of principle, we set the difference frequency to a fixed value, and
load the atoms into a constantly moving lattice. The frequencies of the two dipole
beams are shifted by using two AOMs (see Fig. 3.23). These AOMs are driven
by two phase-locked waveform generators (Agilent N9310, and N5182). Since the
implemented phase stabilization only has a low bandwidth, the relative phase noise
of the two independent waveform generators is unsuppressed at high frequencies
and therefore lowers the lifetime of our trap to 12 ms. Nonetheless, we are able
to demonstrate transportation of atoms over more than 1 mm in each direction
along the nanofiber by loading the atoms into the lattice sites moving at various
velocities (see Fig. 3.23). Up to a velocity of 6 mm/s the number of trapped atoms
does not significantly change but the distribution is elongated in the direction of the
transportation. Higher velocities result in a decrease of the number of transported
atoms, presumably because the efficiency of loading the atoms into the moving
trapping sites is reduced.
However, in principle the loading might be performed with a stationary lattice
and subsequently transported. In this case the transportation efficiency would be
primarily limited by the short lifetime of the trap. A more elaborate phase stabilization should, however, allow us to reduce the atom loss during the conveyance,
and would permit us to trap the atoms along the whole length of the nanofiber.
This would increase the number of trapped atoms by about a factor of up to five,
or perhaps more if the nanofiber length is increased.
Another possible application of such a fiber-optical conveyer belt for atoms might
be the deterministic coupling of atoms with solid state devices as, e.g., micro resonators [118–120] for cavity-QED experiments, or for experiments involving superconducting circuits (Josephson junctions) [7–9]. In this case, the atoms could be
loaded at a distant MOT region and carried to the desired place in the experimental setup. Due to the tight transversal confinement of the trapping fields around
the nanofiber, the atoms can be placed in close vicinity of such a device while
permitting a minimum of interaction with the trapping fields.
3.7
3.7.1
Imaging of the trapped atoms
Fluorescence imaging
For imaging the trapped atoms, we have positioned a home-built microscope objective with a numerical aperture of 0.29 inside the vacuum chamber [101]. This
allows us to collect the fluorescence light, induced by the resonant excitation of the
trapped atoms, and thus create an image on the chip of an electron multiplying
charge coupled device (EMCCD) camera (Andor, DU897).
90
Figure 3.24: (blue curve) Horizontal profile of the fluorescence image of the
nanofiber trap (upper image), the excitation light propagates through the fiber
from the left to the right. The red curve represents the density distribution of
the atoms (see text). Here, the flat central part, indicates maximum filling of
the nanofiber trap. As a reference the fluorescence of the MOT-cloud has been
imaged before loading the trap (green curve) and the scaling has been adjusted
arbitrarily in order to compare the shapes of the curves.
91
For all recorded images the probe laser was detuned by ∆ = −20 MHz with respect
to the AC-Stark shifted D2 (F = 4 → F ′ = 5) transition of cesium and its power
was set to 500 pW, corresponding to twice the saturation intensity at resonance
at the position of the atoms. Such a large detuning was chosen in order to ensure
low attenuation of the probe light along the nanofiber, thereby making all atoms
of the ensemble contributing to fluoresce with similar intensities. The probe light
was pulsed with a pulse length of 2 ms, corresponding to the exposure time of
the EMCCD camera. The power chosen for the probe light ensures low heating
and negligible losses of atoms during the exposure time. The axis of polarization
was aligned perpendicular to the camera view direction in order to maximize the
perceived intensity on the camera chip. All fluorescence images shown in this thesis
were recorded 20 ms after loading the nanofiber trap. This delay time is sufficient
for the atoms to escape from the MOT, thus shielding of the fluorescence light by
the surrounding atom cloud can be neglected. Each shown image is the sum of 320
single background corrected exposures from consecutive experimental runs.
The image in Fig. 3.24 shows the fluorescence of a trapped ensemble of about 2000
cesium atoms over a length of L ≈ 1.2 mm corresponding roughly to the 1/e2 diameter of the cold atom cloud in the MOT. Here, the fluorescence intensity is
proportional to the number density of atoms ρ(z) times the probe light intensity
Iprobe (z), i.e.,
Ifluor. (z) ∝ ρ(z)Iprobe (z)
∝ ρ(z)I0 exp(−σ
Z
z
ρ(z ′ )dz ′ ),
(3.32)
0
where the exponential function accounts for the intensity loss due to absorption
along the nanofiber. The intensity profile of the image (Fig. 3.24, blue curve) can
thus be used to estimate the density distribution ρ(z) of the trapped atoms along the
nanofiber (z-direction). Given that the attenuation of the probe light varies slowly
along z, we can simply approximate Iprobe (z) ≃ I0 exp(−OD/L · z), where OD is
the measured optical density of the atomic sample (OD ≈ 1 for ∆ = −20 MHz).
In Fig. 3.24 the resulting density profile of the trapped atomic ensemble (red curve)
is compared to the profile of the atom cloud in the MOT (green curve). As expected,
the density distributions are consistent in the low density region of the MOT. In
the central part of the MOT, however, the density distribution becomes flat. This
behavior can be explained by assuming that the maximum filling of the nanofiber
trap is limited by the collisional blockade effect [97]. If we assume loading of at most
one atom per lattice site and a filling factor of 0.5 (cf. [97]) for each of the two lattices
above and below the nanofiber, then from the measured density distribution ρ(z) in
PL/∆z
Fig. 3.24 we obtain the number of trapped atoms N = 2 × i=0 ρ(zi )∆z ≈ 2100,
where ∆z = zi − zi−1 ≈ 500 nm is the lattice spacing. The resulting atom number
of 2100 is in good agreement with our absorption measurement performed under
strong saturation (N ≈ 2000) as described in Sect. 3.3.2.
92
3.7.2
Coherent scattering
For the imaging of the trapped atoms relatively high intense excitation pulses
(500 pW) have been used in order to obtain reasonable counting rates on the EMCCD chip. About 60% of the input light power was absorbed by the trapped
atoms (300 pW) and presumably re-emitted into free space. Interestingly, our
camera collected only a tiny fraction of about 1/250 of the power expected under
the assumption of isotropic scattering (1 pW), taking the NA of the objective, the
transmission of the filters, the chip sensitivity and quantum efficiency, the EM gain,
and the timing of the camera into account.
This result would suggest that the fluorescence of the trapped atomic ensemble
displays coherences of the re-emitted light, such that the radiation interferes in a
directed pattern and thus lowers the intensity in the observed solid angle. This
assumption is further confirmed when regarding the optical Bloch equations of a
driven two-level atom subjected to coherent incident light: The coherent fraction
of the scattered intensity is unity in the limit of weak coupling (Ω → 0) (or large
detuning (∆ ≫ Γ)) [122]
coherent
I¯scat.
1 + 4∆2 /Γ2
=
,
1 + 4∆2 /Γ2 + 2(Ω2 /Γ2 )
I¯scat.
(3.33)
where Γ is the atomic decay rate and Ω = deg · E/~ is the Rabi frequency which
quantifies the interaction strength. In our case, this fraction is expected to be about
97%, for an off-resonant saturation parameter of s ≡ 2(Ω2 /Γ2 ) ≈ 2, estimated from
the previously mentioned intensity and detuning of the probe field. Thus, in this
regime the atoms are expected to radiate almost fully coherently1 , like classical
dipoles, with a well defined phase with respect to the excitation field.
The radiation characteristic of such a coherently radiating array of emitters is related to phased arrays in wave theory, which is also the basic principle of directed
antennas. Here, the relative phases of the periodically spaced emitters are aligned
in such a way that the radiation pattern resulting from interference, reinforces
radiation in the desired direction and suppresses it in all other directions [121].
In our case, the atoms are trapped in two one-dimensional arrays with a constant
lattice spacing of ∆z ≈ 500 nm (and a filling factor of 0.5). The excitation via
the fiber guided light imprints a position-dependent phase relation (φ(z) = βz,
see Sect. 1.1) between the atoms, such that the subsequent coherent re-emission
into free space results in a well-defined angular distribution analogous to Bragg
scattering
2π − βprobe ∆z
cos(θmax ) =
,
(3.34)
kprobe ∆z
1
Note that for Zeeman-degenerate transitions spontaneous Raman scattering may increase the
incoherent fraction (up to 30% for Cs [123]). Furthermore, for the case of a large coupled atomic
ensemble the total scattering rate might also be changed [116, 117].
a)
93
b)
Intensity ( ) [a.u.]
1000
100
k
10
probe
1000 100
z
probe
10
1
1
1
1
10
100
1000
10
100
1000
Figure 3.25: (a) Bragg-scattering of the fiber guided probe light due to a
periodic arrangement of trapped atoms. (b) Polar plot of the angular intensity
distribution, simulated for a random filling of 100 atoms in the array (filling factor
of 0.5). Note that the intensity has been scaled logarithmically.
where βprobe is the propagation constant of the fiber-guided mode of the probe light
and kprobe is the corresponding wave vector in vacuum. For an infinitely long and
perfectly periodic lattice this would result in a sharply defined scattering cone with
an opening angle of 2θmax = 2 × 56◦ for the probe light at a wavelength of 852 nm
(see Fig. 3.25(a)).
Due to a random filling of the optical lattice, however, this pattern is expected to
be blurred such that a small part of the coherent radiation is scattered into other
directions. The random distribution of defects in such a √
lattice would result in
a spatial fluctuation of the electric field amplitude ∆E ∝ Nvoids , such that the
intensity of isotropically scattered light would be proportional to the number of
voids in the lattice (Iiso ∝ Nvoids ). In contrast to this, the intensity at the maxima
of the angular distribution is expected to scale quadratically with the number of
2
).
atoms located in the lattice (I(θmax ) ∝ Natoms
Figure 3.25(b) shows a simulated intensity distribution arising from scattering of
such arbitrarily filled lattice sites with 100 atoms and a filling factor of 0.5. The resulting pattern indicates preferred scattering at θ = ±56◦ with additional isotropic
emission, whose intensity is proportional to the number of voids in the lattice. In
our experiment we should be able to observe the scattered power of about half of
the 2000 trapped atoms, but the currently measured power corresponds to only
8 atoms. This result may indicate that the radiation characteristic of the fibercoupled ensemble is altered either due to an unexpected even distribution of atoms
inside the lattice, or probably due to collective radiative effects [117].
94
The investigation of the scattering under the preferred angle might may help to
clarify this circumstance. However, our setup does not provide full optical access
to all angles and we were not able to resolve this pattern of coherent radiation. We
are planning to employ a tunable laser to change the spacing in our optical lattice
via its wavelength, thereby shift the angle of preferred emission into the range of
optical access.
It is worth noting that one special feature might occur in the case of coherent scattering by atoms arranged in a lattice with spacings ∆z < 2π/(βprobe + kprobe ): In
the classical picture, any emitted radiation interferes destructively in all directions
in vacuum if the atoms are closely spaced and coherently excited with a slightly
smaller wavelength than the wavelength in vacuum. As a consequence, deexcitation
of the atoms due to emission into free space will be inhibited. The only remaining decay channel is the nanofiber. By choosing reasonably large trap distances
from the surface the channeling of radiation into the fiber might be strongly suppressed [130]. Hence, the lifetime of the excited atomic states could be substantially
increased. More elaborate calculations by H. Ritsch et al. [117] have shown that in
this case the system may give rise to metastable excitons which propagate in the
lattice without showing a radiative decay.
Chapter 4
Interfacing light and atoms
The key challenge in implementation of practical quantum information technologies
as, e.g., quantum-networks is a faithful mapping of photonic and atomic quantum
states [124]. For this purpose, efficient transfer of quantum states between photons
and atoms might be established by interfacing light and atoms under a strong
light-atom interaction.
As an alternative to the single-atom approach using a high-finesse cavity in the
strong-coupling regime, an ensemble of atoms can provide a similar strong atomlight coupling without a cavity. This approach involves less complexity and provides
a better accessibility to the atoms compared to cavity-QED experiments. Here, the
interaction with a collective mode of the ensemble can be collectively enhanced due
to many-atom constructive interference [23]. As shown in [23, 124] the “effective”
figure-of-merit for the light-atom ensemble interface is given by the resonant optical
density (OD). In this chapter we analyze the coupling of fiber-guided light and the
trapped atomic ensemble in our experiment, and motivate further developments
towards direct integration of laser-cooled atomic ensembles within fiber-networks.
Moreover, this chapter presents three powerful approaches to interface trapped
atoms and fiber-guided light. In the first part the resonant interaction of the
trapped atomic ensemble and the fiber-guided light is outlined. Furthermore, the
spectroscopic properties of the trapped atoms are investigated and the influences
of the ac-Stark shifts and inhomogeneous broadening are discussed.
The second part reports on the observation of fluorescence via the coupling of light
and the guided modes of the nanofiber.
The last part is focused on a non-destructive phase measurement scheme based on
off-resonant light-atom interaction.
96
4.1
Resonant interaction
All the schemes of detection of the trapped atomic ensemble described so far in this
thesis rely on the (near) resonant scattering of light.
The fiber-guided probe light, tuned (close) to the D2 transition of cesium, couples
to the atoms via the evanescent field and gets scattered by the atoms. The response
of the atoms to the near resonant light results in absorption and dispersion of the
probe light, which in turn is perceivable at the output of the fiber. Here, the
presence of a large trapped atomic ensemble results in a strong extinction of the
fiber-guided probe light.
The presented detection scheme takes advantage of the resonant enhancement of
the optical response in conjunction with an efficient coupling of the atoms to the
fiber-guided light. The capacity to establish strong coupling between a photon and
a collective mode of the ensemble makes the presented technique ideally suited for
interfacing photonic and atomic quantum systems.
Method
An important quantity characterizing the probability for absorption and scattering
of photons by atoms is the cross section which can be understood as the effective
area of the atom occluding the incident light beam. In this regard, the efficiency
for coupling fiber-guided light and the atom (or the OD per atom) is given by
OD
,
(4.1)
η = σ/Aeff =
N
where σ is the absorption cross section of the atom which incorporates the dependency on the atomic state as well as on the driven transition, and Aeff = P/I(rat )
represents the effective mode area of the fiber-guided mode [36, 126] (i.e., ratio of
the total power P and the intensity of the evanescent field I(rat )), where rat denotes
the position of the atom. When the distance between the atom and the fiber surface
is small, the light-atom coupling is enhanced by the strong lateral confinement of
the field around the nanofiber, making Aeff comparable to σ [15].
The absorption cross section owing to a dipole transition |nF MF i → |n′ F ′ MF′ ′ i is
related to the atomic transition dipole moment
dnF M,n′ F ′ M ′ = hnF MF |e · d|n′ F ′ MF′ ′ i
(4.2)
and may be expressed by [126]
σ=
ω0
1
|dnF M,n′ F ′ M ′ |2
,
2
2cǫ0 Γ0 ~
1 + 4∆ /Γ20 + s
(4.3)
where e represents the polarization vector of the probe light, d denotes the dipole
operator, ∆ = ω − ω0 is the detuning of the probe light from the (light-shifted)
4.1 Resonant interaction
97
angular transition frequency ω0 , Γ0 is the decay rate of the excited state, s denotes
the saturation parameter (s = I/Isat ), and ǫ0 is the vacuum permittivity. Here
and in the following we use the same nomenclature for the matrix elements and the
quantum numbers as in Sect. 2.2.
In the spherical basis, the absorption cross section for each polarization component
of the probe light (denoted by q) reads
ω0 | hnF kdkn′ F ′ i |2
σ=
2cǫ0 Γ0 ~
F′ 1
F
′
MF ′ q −MF
2
1
1+
4∆2 /Γ20
+s
,
(4.4)
where
J J′ 1
| hnF kdkn F i | = (2F + 1)(2F + 1)
F′ F I
′
′
2
′
2
| hnJkdkn′ J ′ i |2 .
(4.5)
In the upper notation q = ±1 denotes σ ± -polarization, and q = 0 corresponds to
π-polarization. We note that the 3-j symbol in Eq. (4.4) specifies the selection rules
F ′ − F = 0, ±1 and MF′ ′ − MF = q. For the atomic states |nF MF i, defined with
respect to the quantization axis parallel to the dominating electric field in the trap,
i.e., E red , we assume negligible mixing of the excited state MF′ sub-levels due to
ac-Stark shifts (see Sect. 2.2).
The relevant parameters for the D2 transition of cesium (6S1/2 → 6P3/2 ) are:
h6S1/2 kdk6P3/2 i = 3.8 × 10−29 C · m, ω0 = 2π · 351.7 THz, and Γ0 = 5.2 MHz [49].
Note that for the closed sub-transitions |F = 4, M = ±4i ↔ |F = 5, M = ±5i, the
resonant absorption cross section in Eq. 4.4 becomes maximal with σ = 6πc2 /ω02 ,
which is equal to the cross section predicted by the Lorentz model [49].
For probing the atoms, we employ the quasi-linearly polarized HE11 mode as a probe
field. If the polarization axis is aligned orthogonal to the axis containing the atoms
(y-polarization, cf. Fig 4.2(d)), the local polarization of the probe field is purely
linear and tangential to the fiber surface (yielding Aeff (rat ) = 1 × 10−11 m2 , see
Sect. 1.3.2). In the orthogonal orientation of the polarization axis (x-polarization),
however, eprobe is aligned normal to the fiber surface, i.e., the intensity of the
evanescent field becomes maximal at the position of the atoms (Aeff (rat ) = 3.5 ×
10−12 m2 ), but in turn the polarization becomes elliptical (see Sect. 1.3.2). The
polarization orientation in the first case leads to the excitation of σ-transitions
(i.e., MF′ ′ − MF = ±1-transitions are driven simultaneously, with the quantization
axis aligned along Ered (x-direction)), while in the latter case mixed π- and σ ± transitions (i.e., MF′ ′ − MF = 0 and MF′ ′ − MF = ±1) can be excited.
In our experiment the initial state of the atomic ensemble is assumed to be an incoherent mixture of all Zeeman sub-states |F = 4, MF i resulting from the spatially
varying polarization and optical pumping in the MOT. The resonant cross section
98
1.000
x-pol.
y-pol.
0.500
0.100
0.050
0.010
0.005
1.0
1.5
2.0
2.5
3.0
radial distance r/a
Figure 4.1: Resonant coupling efficiency of unperturbed cesium atoms σ/Aeff
to the evanescent light field as a function of the normalized radial distance r/a,
calculated for a fiber radius of a = 250 nm and a wavelength of 852 nm (D2
transition, |F = 4i → |F ′ = 5i). The atomic population is assumed to be equally
distributed among the MF -states. The dashed and the solid line correspond to
the x-polarization and the y-polarization of the probe field, respectively, where the
x-axis aligned parallel to the axis containing the atoms while the y-axis is aligned
perpendicular to it. Note that the ordinate has been scaled logarithmically.
(∆ = 0) according to Eq. (4.4) for unperturbed cesium atoms and an even population of all MF sublevels yields σ = 1.4 × 10−13 m2 , which is found to be independent
of the polarization of the probe light.
For both polarization orientations the radial dependence of σ/Aeff for evenly populated MF -states, are plotted in Fig. 4.1. These plots represent the position dependency of the atom-light coupling in absence of the trapping fields.
Note that due to the MF′ ′ dependent light-shift of the |6P3/2 F ′ MF′ ′ i excited state of
the trapped cesium atoms (see Sect. 2.2), the absorption line of each sub-transition
is shifted differently. The multitude of unequally shifted sub-transitions results in
inhomogeneous broadening of the absorption spectrum and simultaneously leads to
a decrease of the maximal absorbance per atom and the OD for the D2 transition in
cesium. From the maximal values of the absorption cross section σ in the presence of
the trapping fields, calculated for the parameters of our trap, we obtain the reduced
absorption cross sections σy′ = 0.35σ and σx′ = 0.5σ, respectively. Meaning that
at the position of the atoms (rat = 230 nm) each atom would absorb a fraction of
ηy = 0.35 × 1.4 % and ηx = 0.5 × 4 %, respectively, when taking the inhomogeneous
broadening into account.
99
x
y
z
Figure 4.2: (a) Transmission spectrum of the probe beam through the nanofiber
after loading the trap (black squares) for orthogonal, and (b) parallel polarization
(blue points). As a reference, we plotted the spectrum of the MOT-cloud in each
case (green line). The red lines are theoretical fits, see text. (c) Saturation measurement (blue circles) yielding the number of trapped atoms (see Sect. 3.3.2).
(d) Polarizations of the probe light with respect to the atoms used for the measurements in (a) and (b).
4.1.1
Spectroscopy
In order to investigate the spectral properties of the trapped atomic ensemble and
to estimate the effect of inhomogeneous broadening we performed measurements of
the transmission/absorption of the probe light as a function of its detuning.
The measured transmission is calculated according to
T =
(Pat − Pbg )
,
(P0 − Pbg )
(4.6)
where Pat , P0 , and Pbg are the APD signals with atoms, without atoms, and without
probe light, respectively.
For the spectroscopic measurements presented in Fig. 4.2 (a & b) the probe light
power was chosen as about 1 pW. Such a low power ensures that the scattering
rate of the atoms (30 kHz) is smaller than their oscillation frequency in the trap
(≈ 140 kHz), resulting in strongly suppressed recoil heating due to off-resonant
100
raman scattering [127]. This maximizes the number of scattered photons before
atom loss, thereby optimizing the signal. The APD signal is recorded with a digital
storage oscilloscope and averaged over 64 traces. As a reference, the green line in
Fig. 4.2 (a & b) shows the transmission of the probe light versus the detuning with
respect to the D2 (F = 4 → F ′ = 5) transition of cesium without the fiber trap
after abruptly switching off the MOT lasers and magnetic field. In this case, the
measured absorption due to the cold atom cloud around the nanofiber reaches 20%
and the linewidth is only slightly larger than the natural linewidth of cesium due
to the atom-surface interactions [15].
The black squares in Fig. 4.2 (a) show the transmission of the probe light polarized
along the y-axis (σ-polarization) directly after loading the fiber trap. We observe
a strong absorption resulting from a strong increase of the number of atoms in
the evanescent field due to trapping of atoms inside the two-color dipole trap. The
fitted line profile (solid red line) yields a maximum optical depth of OD = 13(2) at a
detuning of 13 MHz and a FWHM of Γ = 20 MHz. The spectrum for π-polarization,
i.e., parallel to E red (x-direction), shows an even higher fitted optical depth of
OD = 32(2) at a detuning of 12 MHz and a FWHM of Γ = 13 MHz (see Fig. 4.2 (b)).
The shift and broadening can be attributed to the state dependent light-shift of the
transition frequency induced by the trapping laser fields (see Sect. 2.2). Compared
to this inhomogeneous broadening due to the various shifts of the excited substates, the broadening due to the thermal motion of the atoms inside the trapping
potential is expected to be negligible (about 0.8 MHz for T = 28 µK).
The transmission spectra in Figs. 4.2 (a & b) are well described by



X |di,j |2q gi,j fi 
T (∆) = exp −OD
,

1 + 4(∆j /Γ0 )2 
(4.7)
i,j
(solid red line), where the exponent accounts for the Lorentzian line profiles corresponding to the transitions between the differently light-shifted new eigenstates,
∆ = ω − ω0 is the detuning of the probe laser frequency with respect to the atomic
LS
resonance frequency in free space ω0 , and ∆j = ∆ − ∆LS
j , where ∆j is the state
dependent light shift induced by the linearly polarized trapping lasers. For convenience we denote the ground state |F MF i by i and the excited state |F ′ MF′ i by j.
Γ0 = 5.2 MHz is the natural linewidth of the cesium D2 transition. The coefficients
gi,j account for the degeneracy, fi for the population, and |di,j |2q for the relative
strength of the transitions (see Eq. (4.3)). They are chosen such that the sum in
the exponent is normalized to one. We assume equally distributed population of the
ground state, i.e., fi = constant. The optical pumping induced by the probe light as
well as collective radiative effects are not included in this model and might account
for the slight discrepancy between the theoretical prediction and the experimental
data.
101
1,0
1,0
0,8
0,8
Transmission
Transmission
0,6
0,4
0,6
0,4
0,2
0,2
0,0
0,0
-40
-20
0
20
Detuning (MHz)
40
-40
-20
0
20
40
Detuning (MHz)
Figure 4.3: Transmission spectra for an atom number of about 100. The polarization of the probe light in (a) is chosen orthogonal to the axis containing
the atoms (y-polarization), and in (b) parallel to it (x-polarization). The blue
dots and black squares indicate the respective measured spectra. As a reference
the respective theoretical spectrum for an even |MF i population is indicated by
a thick solid line. The light blue lines are fitted Lorentzian profiles to the optical
depth OD(∆).
The theoretical transmission spectra according to Eq. (4.7) corresponding to the two
orientations of the probe light (x- and y-polarization) are depicted in Fig. 4.2 (and
Fig. 4.3 for low atom numbers). From the fitted optical depth and the number
of atoms, shown in Fig. 4.2, we infer an absorbance per atom of ηy ≈ 0.6(1) %
and ηx ≈ 1.6(1) %. These values are consistent with our expectation that each
of the trapped atoms at a radial distance of about rat = 230 nm from the fiber
surface would absorb a fraction of σy′ /Aeff (rat )y = 0.5 % and σx′ /Aeff (rat )x = 2 %,
respectively (see above).
4.1.2
Optical pumping
For small numbers of atoms in the trap we are able to resolve the absorbance
profiles, which, in contrast to the expected spectra, are reasonable well described
by Lorentzian line shapes with a FWHM of approximately 13 MHz and 11 MHz for
σ- and π-polarized probe light, respectively (see Fig. 4.3). The low atom number is
obtained by reducing the partial pressure of cesium in our MOT-chamber. For this
measurement we apply probe pulses with a duration of 0.5 ms and a power of about
1 pW, such that each atom scatters only up to 15 photons. In order to minimize
the contribution of heating to the spectrum, each data point corresponding to one
specific detuning is recorded in an individual experimental run.
Despite the fact that the measured spectrum for x-polarization almost resembles the theoretical prediction for an evenly distributed MF -state population (see
102
Fig. 4.3 (b)), the spectrum for y-polarization (blue circles) shows significant deviations from the expected spectral shape calculated for the same population (black
line in Fig. 4.3 (a)). The narrow and less complex structure might be an indication
of a redistribution of the |MF i state population caused by optical pumping.
In particular, the small detuning of the light-shifted resonance, i.e., the position
of the transmission minimum (blue circles in Fig. 4.3 (a)), indicate an increased
admixture of the |F = 4, MF i → |F = 5, MF′ ′ i transitions to the excited state sublevels with a large |MF′ | number (see Sect. 2.2). Such a spectrum would suggest
that resonant scattering of the probe light may give rise to optical pumping of
the ground state sub-level population towards the stretched state |MF = ±4i, as is
expected for cesium vapor under illumination with σ-polarized resonant light [49].
As discussed previously, in the case of polarization axis being aligned parallel to
the axis containing the atoms the probe field excites π- and σ ± -transitions simultaneously, such that optical pumping (out of an even MF population) might be
inhibited. Note that pumping by the MOT cooling light during loading of the trap
is unlikely because the measured spectra of the x- and the y-polarization do not
exhibit the same MF population.
For the y-polarized probe light (σ-polarization) the effect of optical pumping by the
probe light would ultimately lead to a narrowing of the absorption spectrum, which
should eventually converge to a Lorentzian profile with a close to natural linewidth,
after a sufficient number of scattering events [49,128]. However, in the experiment,
this spectral evolution could not be enforced by applying a pump pulse prior to
probing. We suspect that heating due to resonant scattering may counteract the
narrowing effect and may even lead to broadening of the absorption spectra. A
further possibility is that, collective electronic excitations change the damping rate
of the atomic transitions [116], thereby leading to homogeneous broadening of the
probed transition.
Discussion and Outlook
When taking optical pumping into account and assuming the same spectral shapes
for both high and low atom numbers, we would expect an approximately 2 times
larger OD for y-polarization and 1.25 times larger OD for x-polarization than that
estimated for an evenly distributed MF -population. With regard to our attempts
of increasing the OD, optical pumping and state preparation are still under further investigation. For future experiments, preparation of well defined Zeeman
state population may be an important consideration for reducing the influence of
inhomogeneous broadening of the probing transition and thus for increasing the
coupling efficiency. As an example, pumping into the stretched state |MF = ±4i
or the clock state |MF = 0i may even raise the absorbance per atom to ηy = 4.5 %
and ηx = 8 %, resulting in an OD of 90 and 160, respectively, for 2000 atoms.
4.2 Fluorescence of the atomic ensemble
103
96%
2%
2%
MOT
Beams
Figure 4.4: Scheme for excitation of the trapped atoms. The atoms are illuminated by an optical molasses provided by the MOT beams. At the position of the
atoms about two percent of the irradiated power is channeled in each direction of
the fiber.
4.2
Fluorescence of the atomic ensemble
In close proximity to the nanofiber surface, the fluorescence light emitted by the
trapped atoms can be efficiently coupled into the fiber-guided optical mode [39,130].
Due to the strong transverse confinement of the fiber-guided optical mode and the
highly pronounced evanescent field, the probability to emit photons into a nanofiber
becomes substantial for distances approaching the fiber surface.
This particular property provides an alternative for effective detection of the trapped
atoms, and interrogation of the atomic states. Moreover, it has been shown theoretically that the emission of photons from a linear array of atoms into the guided
modes of a nanofiber can be even collectively enhanced, thereby reaching a coupling
efficiency close to unity [129].
For the estimation of the efficiency for the atom to emit into the nanofiber, a
fluorescence measurement has been accomplished by illumination of the trapped
ensemble by the six counter propagating MOT-cooling beams (see Fig. 4.4) and
detection of the fiber-coupled fluorescent light on one or both ends of the TOF.
This approach of excitation has the advantage that expulsion of the atoms from
the trap due to rapid resonant scattering can be diminished, since the photons
are incident from all directions [51]. Moreover, we found that for carefully chosen
detunings and intensities of the cooling light the atoms get additionally laser-cooled
while they are interrogated via the optical molasses, thereby leading to an increased
trap lifetime.
104
Emission rate into the guided modes
The theoretical description of the coupling of fluorescent light to the fiber modes
has been extensively reviewed in [130]. We have employed these results to estimate
the expected coupling efficiency using the expression of the relative emission rate
of atoms in the vicinity of a nanofiber
ηfluo. (r) ≃
γguided (r)
,
γ0
(4.8)
where γguided (r) denotes position dependent spontaneous emission rate into the
guided mode
I(r)
γguided (r) ≃ γ0 γa
,
(4.9)
I(a)
γ0 is the spontaneous emission rate of an atom in free space, and γa is the normalized
decay rate γguided (a)/γ0 at the fiber surface (r = a), and I(r) is the intensity
distribution of the guided mode. According to this model, the coupling efficiency
into the guided modes follows the radial field distribution outside the nanofiber,
i.e, ηfluo. (r) ∝ I(r), as is the case for the atom-light coupling in absorption with
ηabs (r) = σI(r)/Pprobe (see Sect. 4.1).
For a fiber radius of a = 250 nm and a transition wavelength of λ = 852 nm
(D2 line) in conjunction with evenly populated atomic MF sub-levels Ref. [130]
predicts γa ≈ 0.28, thus Eq. (4.8) yields ηfluo. (rat ) = 2.7% at a radial distance
of rat = 230 nm from the fiber surface. Meaning that in our experiment 2.7%
of the fluorescence light is expected to be emitted into the TOF to both sides.
In comparison to that, the resonant absorbance per atom ηabs of the fiber-guided
probe light (isotropic polarization) would yield ηabs (rat ) = 2.7% as well in absence
of inhomogeneous broadening due to the ac-Stark shift. However, in contrast to the
absorption measurements, the coupling efficiency of fluorescence light is expected
not to be influenced by ac-Stark shifts and inhomogeneous broadening.
Experimental realization
The trapped atoms are excited via the MOT-beams and the fluorescence light
coupled to the nanofiber is detected at the output of the TOF using an APD. The
fluorescence signal, recorded during the last seconds of the experimental sequence,
is shown in Fig. 4.5 (black curve).
Before the atoms are transferred from the MOT to the dipole trap (t < 0), we observe fluorescence mainly from the MOT-cloud surrounding the nanofiber [16,17,39]
and from a few trapped atoms. As soon as the loading sequence starts (t = 0), the
detuning is increased in one single step to ∆ = −80 MHz with respect to the light
shifted cooling transition. This reduces scattering in the first place but afterwards
4.2 Fluorescence of the atomic ensemble
105
trap on
trap off
trap on /
repump off
0.7
Fluorescence (pW)
0.6
0.5
0.4
0.3
0.2
repump off
MOT on
MOT
reloading
0.1
0.0
-0.5
0.0
0.5
1.0
1.5
Time (s)
Figure 4.5: Fluorescence excited by the MOT-laser beams and detected as the
light is channeled into the nanofiber waist of the TOF. The loading of the dipole
trap starts at t = 0, and is performed with a fixed detuning of ∆ = −80 MHz.
As a reference the red curve indicates the background level caused by the MOTbeams without the dipole trap. The green curve shows the remaining fluorescence
when the trapped atoms are pumped into a dark state. The boxes on the left and
on the right mark the operation of the MOT.
the accumulation of trapped atoms leads to an increase of the fluorescence signal.
Within the first 200 ms, when about 2000 atoms are loaded into the trap, the measured fluorescence increases up to 0.5 pW, showing a loading characteristic similar
to that measured with our conventional loading sequence (see Sect. 3.3.1). Without
the presence of the trapping fields around the nanofiber the signal drops to the zero
level (red curve). Note that the background level of 0.4 pW corresponding to the
coupling of the MOT-beams into the fiber guided mode has been subtracted for all
plots shown in Fig. 4.5.
As a second reference, the fluorescence is measured after switching off the repump
laser, thereby pumping the atoms into a dark state (green curve). The subsequent
decay into the |F = 3i state decouples the atoms from the cooling light of the MOT
(i.e., the detuning of the probe light increases significantly to 9.2 GHz), and thus
unveils the background level.
Note that the opacity of the MOT-cloud is small at detunings of −80 MHz, hence
the expansion of the atomic cloud during the loading procedure does not affect
the intensity of the MOT-beams arriving at the nanofiber. Thus, the measured
fluorescence is expected to depend only on the number of trapped atoms. The
large detunings also ensure that the losses through reabsorption of the fiber-coupled
light can be neglected. Furthermore, the detuning of the MOT-lasers close to
106
∆ = −80 MHz (with an intensity of about 8 mW/cm2 ), provides molasses cooling
in our trap and allows for counteracting heating and losses of atoms. The resulting
storage time, indicated by the slow exponential decay of the fluorescence signal, is
raised up to τ ≈ 3.5 s, limited by the background pressure of the vacuum chamber.
From the measurements of the maximum fluorescence light power arriving at one
end of the fiber Pmeas. = 0.5 pW, the number of trapped cesium atoms N ≈ 2000 (cf.
Sect. 3.3.2) emitting into the nanofiber, and the total MOT-beams peak intensity
of 8 mW/cm2 , we deduce the coupling efficiency (neglecting losses in the TOF)
η=
Pmeas.
Pscat.
(4.10)
to be η ≈ 2.3(2)%, where the scattering power Pscat. = N Rs ~ω0 is found to be
approximately 22 pW with a scattering rate of Rs ≈ 50 kHz. Here, we have
assumed a saturation intensity of Isat = 2.7 mW/cm2 for isotropically polarized
light [49]. The measured coupling efficiency and the theoretically expected value
ηflou. = 2.7% are in good agreement within the experimental uncertainties. As a
comparison to free-beam optics, the collection efficiency of η ≈ 2.3% would require
an objective lens with an NA of about 0.3 [101]. Note that the depth of focus
(or Rayleigh length) for such an NA would not exceed the length of the ensemble
(> 1 mm) or the nanofiber.
In conclusion, we have experimentally demonstrated efficient probing of fluorescence
emitted by atoms trapped in the vicinity of the nanofiber surface. Such an efficient
coupling of the atomic fluorescence into the TOF, makes the outlined method ideally suitable for investigation of, e.g., spontaneous emission in close vicinity of a
dielectric surface of a wave-guide. Particulary, the strong confinement of the atoms
in the nanofiber trap opens the way for investigation of atom-surface interactions
with high precision and an unprecedented spatial resolution: The possibility to
position the atoms very close to the surface may lead to interesting applications
in terms of cavity quantum electrodynamics (cavity-QED), and studies of van der
Waals and Casimir-Polder interactions with the fiber surface [54–57].
Moreover, for an atomic ensemble prepared in a certain one-excitation state, i.e,
when a single electronic excitation is indistinguishably shared between the atoms,
the coupling efficiency of the subsequently emitted photon into the nanofiber can
be close to unity [129]. This collective enhancement of emission into the nanofiber
might be a key factor in the realization of fiber-coupled single-photon sources, as
required for the implementation of quantum information technologies.
4.3 Nondestructive Phase Detection
4.3
107
Nondestructive Phase Detection
The absorption measurements performed on our system reveal that resonant scattering can be very efficiently employed for interfacing light and matter, making our
system a prime candidate for detection and manipulation of atoms at the quantum
level. A major drawback of this method, however, is the accompanying absorption
and spontaneous re-emission of photons which results in damping of the light-mode
and heating of the probed atomic ensemble.
The successful implementation of quantum information schemes requires means to
make measurements of the atomic ensemble without destroying the internal quantum states being used [131]. Such quantum non-demolition (QND) measurements
are usually made by employing the dispersive effect of the atomic ensemble caused
by the coherent forwards scattering of photons, which has been shown to be more
powerful than absorption or florescence-based detection techniques. Typically, the
dispersion is detected either through interferometry [132], frequency modulation
spectroscopy or phase-contrast imaging, used most often for imaging Bose-Einstein
condensates [138].
In the following we demonstrate that, in the context of the atoms trapped with
an optical nanofiber, a similar QND measurement can be very simply realized by
taking advantage of the intrinsic birefringence of the system. A signal is derived
that is proportional to the optical phase shift induced by the atoms, which is in
turn proportional to the number of atoms trapped around the fiber. In addition
to its non-destructive nature, this technique is fast, simple, features good signal
to noise ratios and avoids the spectral complexity often entailed in the resonant
absorption approach.
Note that parts of this section are prepared for publication [149].
Method
In the nanofiber-based trapping scheme demonstrated above, resonant detection of
the atoms may be achieved by coupling a probe beam at a wavelength of 852 nm
into the fiber. This beam couples efficiently to the atoms via its evanescent field
resulting in an absorbance per atom of the order of one percent (measured at the
light shifted resonance for the strong F = 4 → F ′ = 5 transition in cesium).
For finite detunings such strong coupling also leads to a significant phase shift of the
probe light. This can be understood by considering the Lorentz model which predicts advanced or retarded phase relation between the driving field and the induced
dipole moment depending on the sign of the detuning (see Sect. 2.1). Both the
absorption as well as the phase retardation can
p be derived from the complex polarizability or the complex refractive index ñ = 1 − 4πnα(ω) [137]. For |ñ − 1| ≪ 1
108
and detunings ∆ = ω0 − ω larger than the splitting of the Zeeman sub-levels associated with the ac-Stark shift induced by the trapping fields, we obtain
ñ − 1 =
=
λ X
∆F F ′ + i NF
σI,J,J ′ ,F,F ′ 2
4πl
∆
′
F F ′ + 1 Aeff
F,F
λ3
(2J ′ + 1)
2
X
(2F ′ + 1)
8π
F,F ′
2
∆F F ′ + i NF
J F I
×
,
′
′
F J 1
∆2F F ′ + 1 Aeff l
(4.11)
where σI,J,J ′ ,F,F ′ is the resonant absorption cross-section for each transition labeled
by I, F and J, which are the nuclear, total atomic and total electronic angular
momenta quantum numbers respectively, and where the primed quantum numbers
refer to the excited states. Here, ∆F F ′ = 2(ωF F ′ − ω)/Γ is the normalized detuning
of the probe light with an average wavelength λ, Γ is the natural linewidth, NF is
the population of the ground state, l is the length of the sample, and Aeff is the
effective cross-sectional area of the probe light. The complex response function for
transmitted radiation is given by exp(i(ñ − 1)2π/λ). The absorptive component of
the light-atom coupling can be expressed in terms of the optical density [138]
4πl
Im{ñ − 1}
λ
1
,
= OD0 2
∆F F ′ + 1
OD(∆F F ′ ) =
(4.12)
where OD0 = σJ,J ′ ,F,F ′,I NF /Aeff is the resonant optical density. Similarly, the
phase shift imparted on the probe light is given by
φ(∆F F ′ ) =
=
2πl
Re{ñ − 1}
λ
OD0 ∆F F ′
.
2 ∆2F F ′ + 1
(4.13)
The maximum phase shift, which occurs at ∆F F ′ = 1, is φ0 = OD0 /4. Eqs. (4.12)
and (4.13) show that for significant detunings, the phase shift can be substantial
even when the absorption is negligible, since OD/φ ∝ 1/∆F F ′ . We also note that
for ∆F F ′ ≫ 1, we have φ(∆F F ′ ) = OD0 /2∆F F ′ . A common approach to detect the
phase shift induced by an optically dense atomic medium is to compare the phase of
a probe beam passing through the medium with the phase of an external reference
beam, i.e., by using an interferometer [132,138]. In the case of atoms being trapped
in two linear arrays around the fiber (see Fig. 4.6), one is able to take a similar
approach due to the inherent asymmetry of the coupling between the quasi-linear
HE11 optical modes with linear polarization that propagate in the nanofiber and
the atomic ensemble [32].
600
a
400
200
200
0
y (nm)
400
b
>3
1.0
0
-200
-200
-400
-400
-600
-600 -400 -200 0 200 400 600
x (nm)
<0.1
-600
-600 -400 -200 0 200 400 600
x (nm)
0.3
Intensity
y (nm)
600
109
Figure 4.6: Normalized intensity distribution (I(x, y)/I(rat )max ) scaled by the
maximum intensity of the evanescent field at the position of the atoms I(rat )max
for (a) the polarization axis aligned parallel to the axis containing the atoms and
(b) the polarization aligned perpendicular to it. The filled black circles indicate the
position of the trapped atoms at a distance of 230 nm (black dashed circle) from
the surface of the 500-nm diameter nanofiber (represented by gray cross-section).
The color scheme has been scaled logarithmically.
Here, the rotational symmetry of the propagation medium is broken by the presence
of the arrays of trapped atoms above and below the nanofiber, and the remaining
mirror symmetry dictates that the eigenmodes of the system consist of the HE11
optical mode oriented either parallel or perpendicular to the axis containing the
atoms. We denote these axes the strong-coupling axis and the weak-coupling axis
respectively, because the coupling of the eigenmodes differs significantly due to the
substantial azimuthal variation of the evanescent field intensity of the HE11 mode.
Note that the same feature of the HE11 mode produces the azimuthal confinement
of the trapping potentials [31].
For the probe light at a wavelength of 852 nm the eigenmode with the polarization
axis parallel to the axis containing the atoms, denoted by |Ψk i, couples 2.8 times
more strongly to the atoms than the eigenmode perpendicular to the axis containing
the atoms, |Ψ⊥ i, corresponding to the ratio of intensity at the center of the trap, i.e.,
at a distance of rat ∼ 230 nm from the fiber surface, in each case (see Fig. 4.6). The
resulting difference in coupling leads not only to a higher optical density for |Ψk i
but also results in a larger optical phase shift accumulated along the fiber, thereby
producing a birefringent effect. By probing the atoms with a linear combination
of |Ψk i and |Ψ⊥ i one can measure this differential phase shift by observing the
change it causes to the polarization state. The comparison with the interferometric
method shows here, that the weak coupling polarization mode plays a similar role
as the phase reference and that the probing may therefore be performed with only
one beam, thus avoiding the need for stabilization of a reference beam path.
This method resembles conventional polarization spectroscopy, except that the birefringence does not need to be induced via optical pumping of the atomic sample [92].
110
Figure 4.7: Setup for the birefringent phase measurement. A detuned light
field is coupled into the nanofiber to probe the cesium atoms, which are trapped
in the evanescent field of the nanofiber (zoomed inset). A Stokes measurement
is performed on the outgoing beam using the analyzing waveplates, a polarizing
beam splitter and two avalanche photodiodes.
Figure 4.7 shows a schematic of the measurement approach. A probe beam is coupled into the fiber with the polarization adjusted to 45◦ with respect to the strong
axis in order to excite a superposition of both eigenmodes |Ψk i and |Ψ⊥ i with equal
1
powers, i.e., |Ψin i = √12
in terms of the normalized Jones vector [134].
1
The two initially degenerate modes then propagate through the fiber, interacting
with the birefringent atom-fiber system, thus exiting the other end of the fiber in
a modified polarization state. When the light passes through the trapped atomic
ensemble, the state undergoes a simple transformation of the magnitude and phase
to produce the polarization state at the output of the fiber. This polarization state is
then determined by measuring the Stokes vector by simply splitting the beam with
a polarizing beam splitter (PBS) for different settings of the analyzing waveplates
and measuring the power of either port with an avalanche photodiode (APD). The
resulting polarization of the light incident on the PBS may be expressed in terms
of the Jones calculus formalism:
tk eiφk
0
|Ψout i = R
|Ψin i
0
t⊥ eiφ⊥
1
tk eiφk
= √ R
,
(4.14)
t⊥ eiφ⊥
2
where the light-atom interaction is modeled by transmissions t2k = e−ODk and t2⊥ =
e−OD⊥ and phase shifts φk and φ⊥ , experienced by the basis states |Ψk i and |Ψ⊥ i
respectively, and R is the Jones matrix describing the phase retardation produced
by the analyzing waveplate(s) in Fig. 4.7. Here we assume that the birefringence
of the fiber itself and other optical components is negligible. In any case these
contributions can be compensated by using variable retardation plates as, e.g.,
Berek compensators.
111
Stokes Measurement
By measuring with three different configurations c of the analyzing waveplate(s),
we can fully characterize the beam’s polarization state and obtain the Stokes vector
[135] (normalized here to the total beam power):



1
S0
 S1 
 (Pk,1 − P⊥,1 )/(Pk,1 + P⊥,1 )


S=
 S2  =  (Pk,2 − P⊥,2 )/(Pk,2 + P⊥,2 )
S3
(Pk,3 − P⊥,3 )/(Pk,3 + P⊥,3 )


1


t2k −t2⊥


2
2
tk +t⊥


,

=  2tk t⊥

cos
(φ
−
φ
)
2
2
⊥
k

 tk +t⊥

 2tk t⊥
sin
(φ
−
φ
)
2
2
⊥
k
t +t
k




(4.15)
⊥
The quantities S0 -S3 are the observables of the polarized light field (cf. Fig. 4.8)
via the powers Pk,c and P⊥,c detected at the two output ports of the PBS in three
configurations:
(c = 1) when the analyzer axis is aligned to the strong-coupling axis
1 0
R=
,
0 1
(c = 2) when the analyzer axis is at 45◦ to the strong-coupling axis
1
1 1
R= √
,
2 1 −1
(4.16)
(4.17)
(c = 3) when a quarter-wave plate has been inserted at 45◦ to the analyzer
1 1+i 1−i
R=
,
(4.18)
2 1−i 1+i
respectively, and the total power can of course be measured by the sum of the
PBS powers in any of those configurations. In practice these configurations are
set by a half-waveplate and/or a quarter-waveplate included before the PBS, as
indicated in Fig. 4.7. The second line in Eq. (4.15) is determined by calculating
Pk = |hΨk |Ψout i|2 and P⊥ = |hΨ⊥ |Ψout i|2 from Eq. (4.14) with the appropriate
waveplate matrix R. It is immediately evident from Eq. (4.15) that both S2 and S3
provide the means to measure the phase difference ∆φ = φk − φ⊥ , but S3 is more
convenient because it also gives access to the sign. The prefactor in S3 is unity
when the absorption is negligible (tk = t⊥ = 1), as is the case far off-resonance.
112
Figure 4.8: Poincaré sphere representation of birefringence. The stokes vector
(red arrow) is rotated from the equator at 45◦ pol. position by an angle φ, corresponding to the phase difference induced by the birefringent medium. Without
absorption the length of the Stokes vector Ip , corresponding to the intensity, stays
unchanged. Here, the colored symbols around the sphere indicate the corresponding state of polarization.
It is also worth noting that S1 is independent of the phase difference ∆φ, which
provides a convenient experimental parameter to determine the orientation of the
strong-coupling/weak-coupling axis.
In order to deduce the absolute phase shift in a particular axis, we use the proportionality established in Eq. (4.13) that φ ∝ OD0 ∝ 1/Aeff and therefore proportional
to the local intensity of the light. This assumption is valid as long as the ground
state population is evenly distributed, i.e., the ensemble has not been optically
pumped, as is typically the case for atoms produced by a magneto-optical trap
(MOT) [5]. From the model of the fiber mode, the ratio is then φ⊥ /φk ≃ 1/2.8 (see
Sect. 1.3.2), leading to
φk = (1 − φ⊥ /φk )−1 ∆φ ≃ 1.6 ∆φ.
(4.19)
The method outlined above was applied to the apparatus described in Sect. 3.2.
In order to maintain the linearity of the probe beam polarization under rotation,
a Berek compensator was used to compensate for the birefringence of the beam
splitter used to combine the probe beam with the trapping beams. Berek compensators were also used on the input and output of the fiber to compensate for the
birefringence present in the section of standard optical fiber, e.g., caused by strain
113
due to bending. The result is that the probe beam polarization is linear both at
the input of the TOF and the output after passing the Berek compensator, and
also results in a quasi-linear HE11 mode.
By observing the Rayleigh scattering from the nanofiber, we estimated the impurity
of the polarization to be less than a few percent (this value has been measured at
wavelengths of 780 nm and 1064 nm but we assume it to also hold for the weak
probe beam at a wavelength of 852 nm). Finally, the components of the Stokes
vectors are determined by calculating the ratio of the difference and the sum of the
powers at each port of the PBS as measured with two APDs. To measure S3 , a
quarter-waveplate is included with its optical axis aligned at 45◦ to the PBS.
Without any trapped atoms, the output polarization remains linear. Introducing
trapped atoms will in general change the value of the Stokes measurement, but the
sensitivity will depend on the orientation of the input polarization. By adjusting the
angle of the input polarization (and simultaneously adjusting the output quarterwaveplate to preserve the S3 measurement), one can find the strong-coupling axis
or the weak-coupling axis where the Stokes measurement with atoms goes to 0. The
highest sensitivity to the presence of atoms is obtained in between these axes, i.e., at
∼ 45◦ , corresponding to S3 in Eq. (4.15). The signal near resonance, however, will
be compromised for a number of reasons. Firstly, absorption becomes significant
near resonance, which not only alters the polarization state as described by the
prefactor in S3 in Eq. (4.15), but also results in a loss of signal simply because less
light is available for detection, particularly with the high optical density scenario
examined here. Secondly, the absorptive interaction results in more scattering
events, which begins to perturb the initial preparation of the ensemble via either
optical pumping or heating and thus loss of atoms from the trap. Thirdly, the
spectral shape of the signal is complicated near resonance due to the presence of
the strong trapping fields, which results in a frequency shift and inhomogeneous
broadening of the resonance via ac-Stark shifting of the Zeeman sub-states (see
Sect. 2.2). Furthermore, this inhomogeneous broadening is also different for the
two polarization states, thus complicating the spectrum measured via the phase
difference. Finally, should it be possible to overcome the signal to noise issues
at high optical densities (i.e., where the phase difference ∆φ may exceed π/2),
the spectrum will also exhibit discontinuities due to the wrapping resulting from
the sinusoidal term in the expression for S3 . One may avoid all of these issues,
however, by using the measured signal only at significant detunings, where the
overall absorption is less than a few percent.
Figure 4.9 shows a measurement of the phase shift φk,meas = 1.6 sin−1 (S3,meas ),
where the frequency of the probe has been scanned across the F = 4 → F ′ = 5
transition in cesium. For completeness, we performed a full scan through the resonance, see Fig. 4.9 (light red points). Here, the solitary data points (|∆| > 100 MHz)
have been recorded at a fixed detuning in each individual experimental run. The
114
phase shift (rad)
9
1.6 sin-1 (S3)
6
3
- 100
0
100
200
detuning (MHz)
- 3
- 6
- 9
Figure 4.9: Phase shift in the strong-coupling axis as a function of the detuning
from the F = 4 → F ′ = 5 transition, estimated from the measurement of the
fourth element of the Stokes vector (S3 ). The blue line is a fit using Eq. (4.13).
Note that for the measurement close to the resonance (light red points) optical
pumping as well as inhomogeneous broadening is expected to decrease the phase
shift.
other data points have been taken while scanning the probe light detuning within
5 ms from −80 to +40 MHz. For further estimation of the phase shift we fitted
the measured spectrum using Eq. (4.13). (Note that we have omitted the data
points which have been taken after scanning through the near-resonant region, i.e.,
∆ > −30 MHz.)
The fit in Fig. 4.9 takes the functional form of the unbroadened dispersion in
Eq. (4.13) with the natural linewidth of the transition of Γ = 5.2 MHz. The fitted maximum phase and maximum optical density in the strong-coupling axis are
φk (∆4,5 = 1) = 12(1) and OD0,k = 48(5) respectively.
The measured number of atoms for that measurement was approximately 2000,
yielding an approximate absorbance per atom of η = OD/N = 2.6 % providing an estimate of the effective far-detuned resonant absorption cross-section of
σ = ηAeff = 0.8 × 10−9 cm2 . This is about 40 % less than the expected value of
1.4 × 10−9 cm2 (cf. Sect. 4.1). This may be a result of optical pumping during the
loading process or residual resonant excitation induced by the probe light. However, as expected, we find a larger on-resonance OD than we actually measure in
Abs = 32), simply because far off-resonance dispersion is not afabsorption (OD0,k
fected by inhomogeneous broadening, thus the reduction of the coupling strengths
can be neglected.
115
1.0
1000
0.7
700
0.5
500
0.3
300
0.2
200
Number of Atoms
normalized Phase Shift
150
0.1
20
40
60
80
100
100
Time after Trap Loading (ms)
Figure 4.10: (red curve) Evolution of the phase shift for about 1000 atoms,
P = 4 pW, and ∆ = 165 MHz during 100 ms of continuous probing. As a
reference, the blue dots indicate the loss of atoms in absence of any resonant light,
measured by absorption for different delay times in each individual experimental
run (cf. Sect. 3.3.5).
Non-destructive measurement and influence of the probe light
The capacity of the outlined phase detection scheme to operate far off-resonance,
where absorption is negligible and scattering rates are low, raises the possibility
of performing QND measurements that do not perturb the quantum states of the
trapped atoms. In order to demonstrate the non-destructivity of this measurement
technique, we performed an atom number measurement of the atomic ensemble
using the phase measurement technique over 100 ms and compared it with a series
of conventional absorption measurements with successively longer delay times (see
Fig. 4.10), as we have performed for the lifetime measurements in Sect. 3.3.5.
Due to an yet not fully understood heating mechanism and limitation of the trap
lifetime, the conventional absorption measurement reveals an exponential decay of
the population of the trapped ensemble with a lifetime of about 50 ms. The lifetime
of the phase signal was also measured to be about 50 ms, which indicates that no
significant perturbation in form of heating or optical pumping is produced by the
presence of the probe light, and that the measurement is non-destructive to the
phase shift and the atom number.
In this measurement, the probe light power was about 4 pW, blue-detuned by
165 MHz from the light-shifted F = 4 → F ′ = 5 transition which corresponds
to a scattering rate of about 30 s−1 (or about 3 photons scattered by each atom
over 100 ms measurement. For detunings ∆ . 30 MHz, the lifetime of the phase
measurement drops below that of the absorption measurement, despite there still
being very low absorption. We believe this to be a result of optical pumping by
the probe light that perturbs the initially evenly populated Zeeman sub-states into
116
atomically aligned states [49]. Such a process would alter the relative coupling
of the strong and weak axis, thereby changing the birefringence that provides the
basis for this technique. Indeed, we have observed that under certain conditions
conducive to optical pumping, i.e., at low detuning and high probe light intensity,
the birefringent signal is reduced to almost zero in a few milliseconds.
Conclusions
We have demonstrated the non-destructive phase detection of an atomic ensemble
trapped around an optical nanofiber. The resulting estimate of the effective resonant absorbance per atom of η = 2.6% clearly reveals the high coupling implicit in
this system that makes it an attractive platform for interfacing the quantum properties of an ensemble of cold atoms with optical fiber-based systems. This approach
could be applied to derive information about the quantum state of the atoms, in
order to prepare them in a known state, e.g., via optical pumping. This may also
help to realize a photonic phase gate, in which the phase of a photon is changed
depending on the internal state of the atoms [139].
Chapter 5
Conclusions and Outlook
In this thesis I have presented the realization of a fiber-based optical interface using optically-trapped cesium atoms. I have shown that our method provides an
efficient tool for coupling light and atoms, and that it is ideally suited to interconnecting different quantum systems. Furthermore, it might enable the realization of
fiber-coupled quantum memories in the context of quantum information processing and transmission. The basic principle of the presented coupling scheme relies
on trapping of neutral cesium atoms in a two-color evanescent field surrounding
a nanofiber. The strong confinement of the fiber guided-light, which also projects
outside the nanofiber in the form of an evanescent field, enables strong confinement
of the atoms as well as efficient coupling to resonant light propagating inside the
fiber.
In the theoretical part of my work, I have analyzed the fundamental properties of
light propagation inside an optical nanofiber. The discussed solutions of Maxwell’s
equations provide the exact modeling of the distribution and orientation of the field
outside the fiber, thereby allowing one to compute the trapping potentials and the
atom-light coupling in the vicinity of a nanofiber. The theoretical considerations of
light-atom interaction have been completed by a quantum mechanical description
of the atomic level shifts of the new atomic eigenstates in the presence of two
arbitrarily polarized light fields.
The experimental part of this work reported on the fabrication of tapered optical
fibers (TOF) with a sub-wavelength diameter from standard glass fibers, and their
optimization in terms of transmissivity and persistence against high laser intensities. Our resulting TOFs feature efficient transfer of the guided mode into and
out of the nanofiber waist with negligible losses, and carry up to a few hundred
milliwatts of laser light in vacuum without fusing. This achievement was a crucial
step towards the realization of the nanofiber-based trap.
118
The trapping scheme has been realized by launching a combination of a linearly polarized blue-detuned running wave and a red-detuned standing wave into the TOF,
thereby forming two 1d-lattice potentials above and below the nanofiber. Since this
scheme critically depends on the polarization of the fiber-guided light, a simple procedure of monitoring and controlling of the polarization on the nanofiber waist has
been introduced. Via observation of the angular distribution of Rayleigh scattered
light, emitted by the nanofiber, we are able to precisely align the polarization of
each trapping field.
The implementation of trapping of cesium atoms around an optical nanofiber has
been accomplished using well-established laser-cooling techniques. A conventional
magneto-optical trap (MOT), serving as a source of cold cesium atoms, is overlapped with the waist of the TOF, while the ends of the TOF establish a direct
link to the exterior laser setup. By this means, the cold atoms can be efficiently
transferred from the MOT into the nanofiber trap. Due to the small trapping
volumes the loading operates in the collisional blockade regime resulting in an occupancy of at most one atom per trapping site. Furthermore, our optimized loading
scheme provides sub-Doppler cooling of the atoms to temperatures of about 28 µK,
resulting in a high occupancy of the vibrational ground state in the nanofiber trap.
In addition to the optimization of the loading/cooling process, I have analyzed our
trap in terms of confinement of atoms and their storage time. The measurements
of the trapping frequencies indicate strong confinement of the cesium atoms which
is compatible to the modeled trapping potentials. In contrast to this, the storage
time is significantly shorter than expected. In this context, I have investigated
the impact of several sources of noise which lead to heating and result in the loss
of atoms. Particularly, the phase fluctuations of the standing wave, presumably
induced by spontaneous Brillouin scattering in the TOF, should lead to strong
heating, and is probably the limiting factor of the trap lifetime.
By introducing a moving standing wave in our trap and by imaging of the atoms,
we were able to demonstrate the transportation of the atoms along the nanofiber in
spite of the relatively short trap lifetime in this configuration. This result promises
a viable way of deterministic delivery of individual atoms, and loading of the atoms
along the whole length of the nanofiber.
The successful interfacing of the trapped ensemble and the fiber-guided light is
another central part of this thesis. In the vicinity of the nanofiber surface, each atom
can scatter (absorb) a significant fraction of photons propagating in the nanofiber.
The analysis of the spectroscopic properties of the trapped atoms in absorption
measurements exhibits good agreements with the predictions of the light-induced
ac-Stark shift presented in the theoretical part. With 2000 fiber-coupled atoms
we obtain an optical density (OD) up to 32. We also found signatures of optical
pumping via the probe light, which in this context might be a convenient way to
prepare the atoms in a single Zeeman substate. This should diminish the effects
5.1 Coherent light-matter interaction
119
of inhomogeneous broadening and thus increase the OD. The demonstrated strong
coupling of the atoms and the fiber-guided modes allows us to probe the ensemble
also via a phase measurement even at large detunings where the absorption of the
probe light is negligible. The inherent birefringence, caused by the anisotropy of the
evanescent field and the azimuthal arrangement of the trapped atoms around the
fiber, provides a sensitive means of interrogation of atomic states. In this scheme,
the state of polarization of the probe light changes while propagating through the
nanofiber and interacting with the atomic ensemble. The birefringence induced
by the atoms is then easily detectable via a Stokes vector measurement. At large
detunings from resonance we could even demonstrate the non-destructive nature of
this measurement.
In conclusion, the presented technique for the coupling of light and atoms opens
the route for numerous unique applications, such as the direct integration of atomic
quantum devices into fiber networks, an important prerequisite for large scale quantum communication schemes. More generally, this work is an important step towards the realization of hybrid quantum systems in which atomic ensembles can be
optically interfaced with, e.g., solid state quantum emitters. Trapping and interfacing atoms in close vicinity of solid state devices, coupled via electric or magnetic
interaction, would one allow to combine the advantageous properties of both systems for quantum information processes.
5.1
5.1.1
Coherent light-matter interaction
EIT and quantum memories
In order to proceed towards the implementation of fiber-coupled quantum devices,
it is essential to explore the coherence properties of our system. For this purpose
we are considering the demonstration of electromagnetically induced transparency
(EIT) [140], which provides the possibility of storing quantum information recorded
as a spinwave in an atomic ensemble [141].
The prerequisite for EIT is, besides a very high optical density of the atomic ensemble, the presence of two energy states in which (superposition) the atoms can
remain for a long time without significant loss of coherence. Such is the case
for sub-levels of different angular momentum (spin) within the electronic ground
state of, e.g., cesium atoms exhibiting two metastable lower hyperfine states, e.g,
|gi = |6S1/2 , F = 4i and |si = |6S1/2 , F = 3i, which can be coupled to an excited
state (e.g., |ei = |6P3/2 , F = 4i) either via the control field or the probe pulse, respectively (see Fig. 5.1). Such a Λ-type atomic level structure (see Fig. 5.1) offers
the required long coherence time for the collective spin excitation or a spinwave
(SW), which is required to make the storage and retrieval of photons feasible [141].
120
Transmission (a.u.)
Probe
Ep
1.0
|ei
F =4
Control
c
F=4
=4
9.19 Ghz
|si
F=3
|gi
0.8
0.6
0.4
0.2
0.0
-20
0
20
40
Detuning (MHz)
Figure 5.1: First attempts to realize EIT with fiber-coupled cesium atoms.
(left) A lambda system for cesium making use of the two stable hyperfine states
F = 3 and F = 4, the classical control field denoted by Ωc , and a weak probe
field denoted by Ep . On two-photon resonance, the excitation pathways |si–|ei
and |si–|ei–|gi–|ei are indistinguishable, leading to destructive interference of the
transition amplitude thus causing transparency of the medium. (right) Transmission spectrum of the probe trough the fiber-coupled atoms in the presence of the
coupling field (1 nW). (Note that the transparency window is broadened because
the relative phase of the probe and coupling laser has not been stabilized.)
The proposed deterministic scheme for storing and retrieving photons refers to “slow
light” or “stopped light” which is based on the coherent control of propagation of
the coupled light-matter excitation known as a “dark state polariton” (DSP) [142].
This control is performed under the condition of EIT, where a coherent coupling
field leads to mixing of the photonic (Ê) and excitonic (σ̂spin ) part of the polariton
field [140]
√
Ψ = cos θ Ê − sin θ nσ̂spin ,
(5.1)
with tan2 θ ∝ n/Ω2c , where Ωc denotes the Rabi frequency associated with the coupling field, and n is the atom number density. By adiabatically changing the control
field strength one can thus transfer the coupled excitation into a stationary atomic
spin excitation (spinwave), thereby storing the light pulse in the EIT medium [140].
Depending on the decoherence rate of this medium the light pulse can be retrieved
after a certain storage time.
The coherence times demonstrated in the recent experiments involving atomic ensembles in a MOT are mainly limited by dephasing due to the thermal motion of the
atoms and were found to be less than one millisecond [27]. A Longer coherence time
is feasible by a reduction of the temperature but is still limited due to the free-fall
under gravity. Other approaches employing dipole traps and optical lattices have
shown dephasing effects due to collisions. In comparison to those approaches, our
nanofiber trap stores at most one atom per trapping site and thus prevents atomic
collisions. Furthermore, the thermal motion of the trapped atoms is strongly restricted (almost to the oscillatory ground state), with respect to the propagation
5.1 Coherent light-matter interaction
121
control
quantum
APD
IF
DM
DM
Cs atoms
Figure 5.2: Schematic of a setup for EIT with fiber-coupled atoms. By using
counter propagating control beams stationary DSPs can be created.
direction of the spinwave. Hence, we expect the coherence time provided by our
trap to be limited mainly by electronic excitations and loss of atoms, which might
be in the order of 50 ms. We believe that the fiber-coupled atomic ensemble in our
experiment has the capacity to fulfil the requirements for EIT with fiber-coupled
atoms, which is supported by our first attempts (see Fig. 5.1). Currently, we are
working on the increasing the lifetime and lowering the heating rate in our trap,
which may increase the coherence time to a few hundred milliseconds.
Note that, a further improvement might be obtained by employing a pure blue detuned nanofiber-based trap as we proposed in [40], potentially providing coherence
times of more than a second.
5.1.2
Dark state polariton physics
The targeted improvement of coupling light and the atomic ensemble, e.g., using
optical pumping and loading along the whole nanofiber, may open the route towards efficient photon-photon interaction enabled by EIT [141]. Such non-linear
interaction may not only be a step towards the realization of all-optical quantum
logic gates [147], but would permit to tailor the interaction of dark state polariton
(DSPs).
The enhancement in nonlinear optical efficiency may be produced by either simultaneously slowing down a pair of light pulses in order to enable a long interaction
time [141] or using stationary-light techniques [143]. For the latter, the enhancement can be dramatically increased via a Kerr-like interaction [143,144]. In this case
the probability of interaction between two single photons scales as ∼ ODλ2 /Aeff ,
resulting in a significant phase shift for the photons. The required optical depth
OD > 100 can be realized within modest improvements (see Sect. 4.1), thus nonlinear interaction of even two guided photons appears within reach.
122
Finally, the demonstrated coupling scheme opens up the possibility of creating
strongly interacting many-body photon states, such as DSPs which can be associated with massive quasi-particles of variable masses. Moreover, due to the strong
photon-photon interaction it is feasible to reach the Tonks-Giradeau regime where a
strong confinement of bosonic DSPs leads to “fermionization” and “crystallization”
of light [145]. These kinds of applications may give new insights into the physics of
strongly correlated systems and may be an exciting prospect for quantum simulation of matter Hamiltonians in 1D using quantum optical systems [148].
List of Figures
1.1
Geometry of a standard glass fiber . . . . . . . . . . . . . . . . . . .
6
1.2
Plot of the Bessel functions of the first and second kind . . . . . . .
8
1.3
Plot of the HE and EH branches . . . . . . . . . . . . . . . . . . . .
12
1.4
Plot of the TE and TM branches . . . . . . . . . . . . . . . . . . . .
13
1.5
Normalized propagation constant β/k as a function of V for a few
of the lowest-order modes of a step-index fiber [34], where k denotes
the vacuum wavenumber. . . . . . . . . . . . . . . . . . . . . . . . .
14
1.6
Vectorial plot of the transverse field components for σ + polarized light 15
1.7
Ellipticity ε = |Er |/|Eφ | of the polarization of the electric fields versus the radial distance r/a. . . . . . . . . . . . . . . . . . . . . . . .
16
Intensity distribution of the HE11 mode for rotating polarization in
the transverse plane . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
Radial intensity distribution of the cylindrical-coordinate components of the HE11 mode for rotating polarization . . . . . . . . . . .
18
1.8
1.9
1.10 Radial intensity distribution of the HE11 mode for linear polarization 21
1.11 Vectorial plot of the transverse electric field components . . . . . . .
23
1.12 Intensity distribution of the HE11 mode for linear polarization in the
transverse plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
1.13 Maximal intensity of the evanescent field as a function of the fiber
radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
1.14 Schematic of a tapered optical fiber (TOF) . . . . . . . . . . . . . .
26
1.15 Schematic of the pulling rig . . . . . . . . . . . . . . . . . . . . . . .
27
1.16 Optical microscope and SEM pictures of a tapered fiber . . . . . . .
28
124
List of Figures
1.17 Transmission of laser light through a TOF during the pulling procedure 29
2.1
Polarizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
2.2
AC-Stark shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
2.3
Optical potential of S1/2 , F = 4 state . . . . . . . . . . . . . . . . . .
44
2.4
Level diagram of Cesium . . . . . . . . . . . . . . . . . . . . . . . . .
46
2.5
Optical potential of |6P3/2 , F = 5i state . . . . . . . . . . . . . . . .
46
2.6
AC-Stark shift of the 6S1/2 , F = 4 ground state and the 6P3/2 , F = 5
excited state of atomic cesium. . . . . . . . . . . . . . . . . . . . . .
47
3.1
Doppler cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
3.2
Magneto Optical Trap . . . . . . . . . . . . . . . . . . . . . . . . . .
52
3.3
Level-scheme of the Cs atom . . . . . . . . . . . . . . . . . . . . . .
54
3.4
Vacuum chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
3.5
Magneto Optical Trap . . . . . . . . . . . . . . . . . . . . . . . . . .
57
3.6
Experimental setup of the fiber-based atom trap . . . . . . . . . . .
59
3.7
Timing sequence of the experiment . . . . . . . . . . . . . . . . . . .
61
3.8
Loading curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
3.9
Saturation measurement . . . . . . . . . . . . . . . . . . . . . . . . .
63
3.10 Polarization control with Rayleigh scattered light . . . . . . . . . . .
65
3.11 Imaged Rayleigh scattered light of the nanofiber waist . . . . . . . .
66
3.12 Determination of the trapping frequencies . . . . . . . . . . . . . . .
67
3.13 Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
3.14 Adiabatic lowering . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
3.15 Numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
3.16 Temperature measurement . . . . . . . . . . . . . . . . . . . . . . . .
75
3.17 Heating rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
3.18 Heating by dipole force fluctuation . . . . . . . . . . . . . . . . . . .
78
3.19 Laser intensity noise . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
3.20 Setup for measurement of the phase noise . . . . . . . . . . . . . . .
83
List of Figures
125
3.21 Phase noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
3.22 Conveying of atoms
. . . . . . . . . . . . . . . . . . . . . . . . . . .
87
3.23 Conveying of atoms
. . . . . . . . . . . . . . . . . . . . . . . . . . .
88
3.24 Line scan of the fluorescence image . . . . . . . . . . . . . . . . . . .
90
3.25 Bragg scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
4.1
Resonant coupling efficiency . . . . . . . . . . . . . . . . . . . . . . .
98
4.2
Transmission spectra of the probe beam through the nanofiber after
loading the trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
4.3
Low-absorbance spectra . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.4
Emission into the guided modes . . . . . . . . . . . . . . . . . . . . . 103
4.5
Fluorescence
4.6
Transverse intensity distribution I(x, y) . . . . . . . . . . . . . . . . 109
4.7
Setup for the birefringent phase measurement . . . . . . . . . . . . . 110
4.8
Poincaré sphere representation of birefringence . . . . . . . . . . . . 112
4.9
Phase Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.10 Time evolution of the phase . . . . . . . . . . . . . . . . . . . . . . . 115
5.1
First attempts to realize EIT with fiber-coupled cesium atoms . . . . 120
5.2
Setup for EIT with fiber-coupled atoms . . . . . . . . . . . . . . . . 121
List of Tables
3.1
List of elements labeled in Fig. 3.5. . . . . . . . . . . . . . . . . . . .
58
3.2
List of heating effects and the corresponding heating rates. . . . . .
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Danksagung
An dieser Stelle möchte ich Allen danken die am gelingen dieser Arbeit beteiligt
waren.
Mein erster Dank gilt meinem Betreuer, der es mir ermöglicht hat, an diesem interessanten Projekt mitwirken zu dürfen, und für die Unterstützung, die vielen
Ratschläge und wertvollen Diskussionen die meinen Einblick in die Quantenoptik
vertieft haben.
Allen Mitglieder der Arbeitsgruppe möchte für die super nette Atmosphäre und die
große Hilfsbereitschaft danken. Ganz besonderer Dank gebührt den Kollegen die
an meiner Seite so manch lange Messnacht verbracht haben: Darunter Guillem, mit
dessen Hilfe das Experiment verwirklicht werden konnte, der mir auch abseits der
Physik ein geschätzter Freund geworden ist. Regine und Daniel die darauf unser
Team verstärkt haben und durch ihren wertvollen Einsatz wesentlich zum Erfolg
des Projekts beigetragen haben. Unseren PostDoc Sam, der mit seinem breiten
Wissen uns viele neue Anregungen bereitet, und den experimentellen Ablauf immens vereinfacht hat. Dem neuen Doktoranden Rudolf danke ich für das Auffinden
und Überprüfen von meinen fehlerhaften Berechnungen mit Mathematica. Auch
danke ich allen die die an der Korrektur meiner Arbeit mitgewirkt haben. Danken
möchte ich auch meinen ehemaligen Kollegen, insbesondere Dietmar, Wolfgang und
Florian für die Unterweisung in die experimentelle Arbeit sowie für die Realisierung
der Faserziehanlage ohne die diese Arbeit nicht verwirklicht hätte werden können.
Insbesondere gilt mein Dank der Administrative des Instituts; dem Sekretariat
mit Christine und Elvira, der Etatverwaltung, Warenannahme, usw., dafür dass
sie mit der reibungslose Abwicklung aller Verwaltungaufgaben uns permanent den
Rücken frei halten. Für die technische Unterstützung danke ich Herrn Lenk sowie
Michael für die viele Hilfe in Elektronik Angelegenheiten, sowie den mechanischen
Werkstätten des Instituts, die durch ihre kompetente und hochqualifizierte Arbeit
zum Erfolg der Arbeit entscheidend beigetragen haben.
Ganz besonders danken möchte ich all denen Menschen die mich auch in anderen
Belangen meines Lebens unterstützen. Meiner Familie die immer für mich da war
und immer an mich geglaubt haben. Meiner Freundin Regine danke ich für die
Unterstützung und Rückhalt die sie mir auch in schwierigen Zeiten gegeben hat.
Danke für die Geduld mit mir!
Allen Kollegen und Freunden die hier aus Platzgründen unerwähnt bleiben wünsche
ich alles Gute.

Optical Interface Based on a Nanofiber Atom-Trap

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